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A101686
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a(n) = Product_{i=1..n} (i^2 + 1).
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28
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1, 2, 10, 100, 1700, 44200, 1635400, 81770000, 5315050000, 435834100000, 44019244100000, 5370347780200000, 778700428129000000, 132379072781930000000, 26078677338040210000000, 5893781078397087460000000, 1514701737148051477220000000
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OFFSET
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0,2
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COMMENTS
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Sum of all coefficients in Product_{k=0..n} (x + k^2).
Row sums of triangle of central factorial numbers (A008955).
"HANOWA" is a matrix whose eigenvalues lie on a vertical line. It is an N X N matrix with 2 X 2 blocks with identity matrices in the upper left and lower right blocks and diagonal matrices containing the first N integers in the upper right and lower left blocks. In MATLAB, the following code generates the sequence... for n=0:2:TERMS*2 det(gallery('hanowa',n)) end. - Paul Max Payton, Mar 31 2005
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REFERENCES
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Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 559-561, Section 147. - N. J. A. Sloane, May 29 2014
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LINKS
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FORMULA
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G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^2*x). - Paul D. Hanna, Jan 07 2013
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - ((k+1)^2+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = Gamma(n+1+i)*Gamma(n+1-i)*sinh(Pi)/Pi.
a(n) ~ 2*exp(-2*n)*n^(2*n+1)*sinh(Pi).
G.f. for 1/a(n): hypergeom([1], [1-i, 1+i], x).
E.g.f. for a(n)/n!: hypergeom([1-i, 1+i], [1], x), where i=sqrt(-1).
D-finite with recurrence: a(0) = 1, a(n) = (n^2+1)*a(n-1). (End)
a(n+3)/a(n+2) - 2 a(n+2)/a(n+1) + a(n+1)/a(n) = 2. - Robert Israel, Oct 25 2015
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MAPLE
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p := n -> mul(x^2+1, x=0..n):
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MATHEMATICA
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Table[Product[k^2+1, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 11 2013 *)
Table[Pochhammer[I, n + 1] Pochhammer[-I, n + 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)
Table[Abs[Pochhammer[1 + I, n]]^2, {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2016 *)
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PROG
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(PARI) {a(n)=if(n==0, 1, 1-polcoeff(sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, 1+j^2*x+x*O(x^n))), n))} \\ Paul D. Hanna, Jan 07 2013
(Python)
from math import prod
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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