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To begin

We have 29 surahs beginning with the initial letters. Only 14 letter of the 29 letters of the alphabet (counting Alif and Hamza as separate letters) are used. They are:
 
Alif13
Ha7
Ra6
Sin5
Sad3
Taa4
Ain2
Qaf2
Kaf1
Lam13
Mim17
Nun1
Haa2
Ya2
Total 78

 

Qaf -

Using Qaf as a starting point seems most logical as it provides a moderately easy starting point. Many counters have found the 57 qafs in Surah Qaf, and also the 57 in the only other surah with Qaf as one of its initial letter, Surah 42, Ash Shura.

Counting the qafs in Surah 50 gives 57 qafs. 34 of the 45 lines/ayat contain qafs.

Counting the qafs in Surah 42 gives 57 qafs. 31 of the 53 lines/ayat contain qafs.

In the hafs version of the printed text in Surah 42, there are a few occurrences of extra words added in small letters above the text, for example in ayah/line 5, there is a high-qaf-lam-alif-maksura. Counting the Qafs in these superscripts gives a total qaf count of 71.

From this, we may observe that as the superscripts are not part of the oral text, they are not what we are counting.

Additionally, Surah 42 has another 4 initial letters. The counts for each are as follows:

Ha    51
Mim    295
Ayn    98
Sin    53

This may point us to the fact that we are looking for totals spanning surahs rather than total in individual surahs. The fact that there are 57 in both Surah 50 and 42 are merely a device to point us on our way. The total of 114 (equalling the number of Surahs) may be more significant, or maybe it is just another signpost.

It also raises a question for Surah 42: Why are Thal (total 57), Shin (total 38), not used as initial letters if it is mere counts we are looking for?

At this point, this is as much as may be gained from counting Qafs. We must move onto the other single letter surahs, beginning with Surah 68.

Nun

Counting the Nuns in Surah 68 gave a total of 131. I was expecting 132 as Khalifa had spelled out the initial Nun, i.e. nun.wa.nun, to take his total from 132 to 133. I seem to be missing a nun. However, I have eyeballed the Surah, and cannot find the purportedly missing nun.

Here are my counts per line/ayah. If anyone can point me to where I might have miscounted, my email is at the top of the page.

line	nuns	line	nuns	line	nuns	line	nuns	line	nuns	line	nuns
1	2	11	1	21	2	31	4	41	3	51	7
2	4	12	1	22	4	32	7	42	3	52	1
3	3	13	1	23	2	33	2	43	3		
4	1	14	4	24	3	34	5	44	5		
5	1	15	2	25	1	35	3	45	2		
6	1	16	1	26	2	36	1	46	2		
7	4	17	6	27	3	37	1	47	2		
8	1	18	2	28	1	38	2	48	2		
9	3	19	3	29	5	39	4	49	4		
10	1	20	0	30	1	40	0	50	2		

 

Surah 19

Surah 19 begins with the initial letters of kaf, haa, ya, ayn and sad. The Kaf is the only occurrence of Kaf as an initial letter.

Here are the counts:

Kaf    137
Haa    148
Ya    248
Ayn    117
Sad    26

Surah 42 contained the only other Ayn used as an initial. Totalling the Ayns gives 98 (from surah 42) + 117 (From Surah 19) = 215.

Khalifa counts

Here is where I concur (blue) and differ (red) with Khalifa's counts. Black means not yet counted. Much work remains to decide which counts are correct or incorrect, or even if my own match the hafs printed version.
 
Table 11: Parameters of the 14 Individual Quranic Initials
Initial Sura , Verse, & (Frequency)
of Initial in Each Sura Sura
First
Sura
Last
Sura
A (Alef) 2:1 (4502), 3:1 (2521), 7:1 (2529),
10:1 (1319), 11:1 (1370), 12:1 (1306),
13:1 (605), 14:1 (585), 15:1 (493),
29:1 (774), 30:1 (544), 31:1 (347),
32:1 (257)

 

2 32
L (Laam) 2:1 (3202), 3:1 (1892), 7:1 (1530),
10:1 (913), 11:1 (794), 12:1 (812),
13:1 (480), 14:1 (452), 15:1 (323),
29:1 (554), 30:1 (393), 31:1 (297),
32:1 (155)

 

2 32
M (Mim)

 

2:1 (2195), 3:1 (1249), 7:1 (1164),
13:1 (260), 26:1 (484), 28:1 (460),
29:1 (344), 30:1 (317), 31:1 (173),
32:1 (158), 40:1 (380), 41:1 (276),
42:1 (
300), 43:1 (324), 44:1 (150),
45:1 (200), 46:1 (225)
2 46
S (Saad) 7:1 (97), 19:1 (26), 38:1 (29) 7 38
R (Ra)

 

10:1 (257), 11:1 (325), 12:1 (257),
13:1 (137), 14:1 (160), 15:1 (96)
10 15
K (Kaf) 19:1 (137) 19 19
H (Ha) 19:1 (175), 20:1 (251) 19 20
Y (Ya) 19:1 (343), 36:1 (237) 19 36
`A(`Ayn) 19:1 (117), 42:2 (98) 19 42
T (Ta)

 

20:1 (28), 26:1 (33), 27:1 (27),
28:1 (19)
20 28
S (Seen)

 

26:1 (94), 27:1 (94), 28:1 (102),
36:1 (48), 42:2 (
54)
26 42
H (HHa)

 

40:1 (64), 41:1 (48), 42:1 (53),
43:1 (44), 44:1 (16), 45:1 (31),
46:1 (36)
40 46
Q (Qaf) 42:2 (57), 50:1 (57) 42 50
N (NuN) 68:1 (133) 68 68
  43423 295 514

Grand Total = 43423+295+514 = 44232 = 19x2328