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%I #7 Aug 18 2015 00:20:58
%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,4,9,10,5,1,1,6,10,16,15,6,1,1,5,18,
%T 20,25,21,7,1,1,8,15,40,35,36,28,8,1,1,9,27,35,75,56,49,36,9,1
%N A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).
%C The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.
%e Row six begins 1 6 18 40 75 126 ... because rows two and three are
%e 1 2 3 4 5 6 ...
%e 1 3 6 10 15 21 ...
%e The array begins
%e 1 1 1 1 1 1 1 1 1 A000012
%e 1 2 3 4 5 6 7 8 9 A000027
%e 1 3 6 10 15 21 28 36 45 A000217
%e 1 4 9 16 25 36 49 64 81 A000290
%e 1 4 10 20 35 56 84 120 165 A000292
%e 1 6 18 40 75 126 196 288 405 A002411
%e 1 5 15 35 70 126 210 330 495 A000332
%e 1 8 27 64 125 216 343 512 729 A000578
%e 1 9 36 100 225 441 784 1296 2025 A000537
%e 1 8 30 80 175 336 588 960 1485 A002417
%e 1 6 21 56 126 252 462 792 1287 A000389
%e 1 12 54 160 375 756 1372 2304 3645 A019582
%e 1 7 28 84 210 462 924 1716 3003 A000579
%e 1 10 45 140 350 756 1470 2640 4455 A027800
%e 1 12 60 200 525 1176 2352 4320 7425 A004302
%e 1 16 81 256 625 1296 2401 4096 6561 A000583
%e 1 8 36 120 330 792 1716 3432 6435 A000580
%e 1 18 108 400 1125 2646 5488 10368 18225 A019584
%e 1 9 45 165 495 1287 3003 6435 12870 A000581
%e 1 16 90 320 875 2016 4116 7680 13365 A119771
%e 1 15 90 350 1050 2646 5880 11880 22275 A001297
%e 1 12 63 224 630 1512 3234 6336 11583 A027810
%e 1 10 55 220 715 2002 5005 11440 24310 A000582
%e 1 24 162 640 1875 4536 9604 18432 32805 A019583
%e 1 16 100 400 1225 3136 7056 14400 27225 A001249
%e 1 14 84 336 1050 2772 6468 13728 27027 A027818
%e 1 27 216 1000 3375 9261 21952 46656 91125 A059827
%e 1 20 135 560 1750 4536 10290 21120 40095 A085284
%p A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # _R. J. Mathar_, Sep 09 2009
%Y Cf. A000040 A007318.
%Y Cf. A064553 (second diagonal), A080688 (second diagonal resorted).
%K easy,nonn,tabl
%O 1,5
%A _Alford Arnold_, Mar 29 2007
%E A-number added to each row of the examples by _R. J. Mathar_, Sep 09 2009
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