A000457 Exponential generating function: (1+3*x)/(1-2*x)^(7/2). (Formerly M4736 N2028)

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

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 *)

Prog

(PARI) for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ G. C. Greubel, May 15 2018
(Magma) [Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // G. C. Greubel, May 15 2018

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.

Sequence in context: A123512 A079515 A024131 * A240681 A113348 A193274

Adjacent sequences: A000454 A000455 A000456 * A000458 A000459 A000460

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Sascha Kurz, Aug 15 2002

Status

approved