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A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

%I A140729 #8 Feb 16 2025 08:33:08
%S A140729 40,2100,324000,117771500,86640153600,115851776040000,
%T A140729 260111401804800000,922852527136155000000,4931966428685936640000000,
%U A140729 38193820496218904209973280000,415101787718859995456102400000000
%N A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.
%C A140729 The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:
%C A140729 ===================================================================
%C A140729 ..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|
%C A140729 k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047
%C A140729 k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|
%C A140729 k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|
%C A140729 k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|
%C A140729 k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|
%C A140729 k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|
%C A140729 ===================================================================
%H A140729 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>
%F A140729 A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].
%F A140729 a(n) ~ Pi^(3/2) * n^(4*n + 1/2) / (2^(n - 3/2) * 3^(n-1) * exp(3*n+2)) * (1 + (3*log(n) + 3*gamma + 5/4)/n), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Aug 29 2023
%e A140729 a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
%e A140729 a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
%e A140729 a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
%e A140729 a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
%e A140729 a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
%e A140729 a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.
%p A140729 A130729 := proc(n) n!*(n+1)!*(n-2)^n*pochhammer(1+(5-n)/(n-2),n)/6^n ; end: seq(A130729(n),n=3..30) ; # _R. J. Mathar_, May 31 2008
%Y A140729 Cf. A000292, A000330, A002411, A002412, A002413, A002414, A140729.
%K A140729 easy,nonn
%O A140729 3,1
%A A140729 _Jonathan Vos Post_, May 25 2008
%E A140729 More terms from _R. J. Mathar_, May 31 2008