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

40, 2100, 324000, 117771500, 86640153600, 115851776040000, 260111401804800000, 922852527136155000000, 4931966428685936640000000, 38193820496218904209973280000, 415101787718859995456102400000000

Offset: 3

Author

Jonathan Vos Post, May 25 2008

Keywords

• easy
• nonn

Comments

The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:

===================================================================

..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|

k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047

k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|

k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|

k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|

k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|

k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|

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Examples

			a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.
		

Links

• Eric W. Weisstein, Pyramidal Number

Crossrefs

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A140729.

Programs

Maple

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

Formula

A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].

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

Extensions

More terms from R. J. Mathar, May 31 2008