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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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.

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

EXAMPLE

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

MATHEMATICA

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