A003235 -id:A003235

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A003236

a(n) = Sum_{k=0..n} (-1)^(n-k) C(n,k)*C((k+1)^2, n).
(Formerly M3107)

+10
3

1, 3, 24, 320, 6122, 153762, 4794664, 178788528, 7762727196, 384733667780, 21434922419504, 1326212860090560, 90227121642144424, 6694736236093168200, 538028902298395832832, 46558260925421295229568, 4316186393637505403773328

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

OFFSET

0,2

REFERENCES

H. W. Gould, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..300

Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.

FORMULA

a(n) ~ c * d^n * (n-1)!, where d = 4 / (w*(2-w)) = 6.17655460948348035823168... and c = exp(1/2 - w^2/8) / (Pi*sqrt(2*w*(1-w))) = 0.740112385268663459927202070799244309431121698475089032623558890186368006364..., where w = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Dec 13 2020, updated Jul 09 2021

a(n) / A003235(n) ~ -2 / LambertW(-2*exp(-2)) = 4.92155363456750509... - Vaclav Kotesovec, Jul 09 2021

MATHEMATICA

Table[Sum[(-1)^(n-k) * Binomial[n, k] * Binomial[(k+1)^2, n], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 13 2020 *)

CROSSREFS

Cf. A346183.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Sean A. Irvine, Mar 19 2015

STATUS

approved

A346184

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(k^2, n).

+10
1

1, 1, 6, 96, 2330, 76230, 3132192, 154830704, 8942749020, 590880389676, 43950871549640, 3634094909879808, 330648849617038680, 32827596801363717080, 3531510395923598074560, 409199784951469138012800, 50807611780916913209679632, 6729703201077108496483268880

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..17.

FORMULA

a(n) ~ 2^(2*n - 1/2) * n^(n - 1/2) / (sqrt(Pi*(1+c)) * exp(n + (2+c)^2/8) * (c*(2+c))^n), where c = LambertW(2*exp(-2)) = 0.21771510575709011079475830443...

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[k^2, n], {k, 0, n}], {n, 0, 20}]