A000457 Exponential generating function: (1+3*x)/(1-2*x)^(7/2).

1, 10, 105, 1260, 17325, 270270, 4729725, 91891800, 1964187225, 45831035250, 1159525191825, 31623414322500, 924984868933125, 28887988983603750, 959493919812553125, 33774185977401870000, 1255977541034632040625

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			G.f. = 1 + 10*x + 105*x^2 + 1260*x^3 + 17325*x^4 + 270270*x^5 + ... - _Michael Somos_, Dec 15 2023
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
• F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
• C. Jordan, Calculus of Finite Differences. Eggenberger, Budapest and Röttig-Romwalter, Sopron 1939; Chelsea, NY, 1965, p. 172.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200
• Selden Crary, Richard Diehl Martinez, and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
• H. W. Gould, Harris Kwong, and Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
• C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.
• Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
• J. Riordan, Notes to N. J. A. Sloane, Jul. 1968
• Eric Weisstein's World of Mathematics, Stirling Number of the First Kind.

Crossrefs

Equals (1/2)*A000906.

Third column of triangle A001497.

Second column (m=1) of unsigned Laguerre-Sonin a=1/2 triangle |A130757|.

Diagonal k=n-1 of triangle A134991.

Cf. A160473, A163939.

Programs

Magma

[Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // G. C. Greubel, May 15 2018

Mathematica

Table[(2n+3)!/(3!*n!*2^n), {n,0,30}] (* G. C. Greubel, May 15 2018 *)

PARI

for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ G. C. Greubel, May 15 2018

Formula

a(n) = (2n+3)!/( 3!*n!*2^n ).

a(n) = (n+1)*(2*n+3)!!/3, n>=0, with (2*n+3)!! = A001147(n+2).

a(n) = Sum_{j=0..n} (j + 1) * Eulerian2(n + 2, n - j). - Peter Luschny, Feb 13 2023

Extensions

More terms from Sascha Kurz, Aug 15 2002