A047974 -id:A047974 - cp's OEIS frontend

A064571 Binomial transform of (3n)!.

1, 7, 733, 365059, 480457465, 1310073012031, 6410226944228437, 51135786265589035963, 620857308610340190325489, 10894455325845833715898309495, 265361776433005578042408841530061, 8686235999260031219427964764144394867, 372097544110411566152076374938114937481193

Offset: 0

Karol A. Penson, Sep 20 2001

From Peter Bala, Mar 20 2022: (Start)
The congruence a(n+k) == a(n) (mod k) holds for all n and k.
It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, taken modulo 5 the sequence becomes [2, 3, 4, 0, 1, 2, 3, 4, 0, 1, ...], a purely periodic sequence with period 5.
More generally, the same property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1 (see Bala, Theorem 1). (End)

Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Cf. A000522, A064570.

Table[Sum[Binomial[n, k] * (3*k)!, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Oct 30 2017 *)

for(n=0,12,print1(round(intnum(x=0,999,exp(-x)*(1+x^3)^n)),", ")) - Gerald McGarvey, Oct 12 2007

In Maple notation: a(n) = hypergeom([1, 1/3, 2/3, -n], [], -27), n = 0, 1, ....
a(n) = Integral_{x = 0..infinity} (1+x^3)^n*exp(-x) dx. - Gerald McGarvey, Oct 12 2007
From Vaclav Kotesovec, Oct 30 2017: (Start)
a(n) = (27*n^3 - 27*n^2 + 6*n + 1)*a(n-1) - 3*(n-1)*(27*n^2 - 45*n + 20)*a(n-2) + 27*(n-2)*(n-1)*(3*n - 4)*a(n-3) - 27*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ sqrt(2*Pi) * (3*n)^(3*n + 1/2) / exp(3*n). (End)
From Peter Bala, Mar 20 2022: (Start)
a(n) = Sum_{k = 0..n} binomial(n,k)*(3*k)!.
a(n) = 1 + 6*n*(1 + 60*(n-1)*(1 + 168*(n-2)*(1 + 330*(n-3)*(1 + 546*(n-4)*(1 + ...*(1 + (3*(3*m+1)*(3*m+2))*(n-m)*(1 + ...))))))).
a(n) == 1 (mod 6).
O.g.f: Sum_{k >= 0} (3*k)!*x^k/(1 - x)^(k+1).
E.g.f.: exp(x)*Sum_{k >= 0} (3*k)!/k!*x^k. (End)

Corrected and extended by N. J. A. Sloane, Oct 29 2006

A112243 Expansion of exp(x(1+x)/(1-2x)).

Original entry on oeis.org

1, 1, 7, 55, 577, 7441, 113671, 2003527, 39971905, 889608097, 21834577351, 585555975511, 17027451783937, 533460597334705, 17908302027585607, 641152804988733031, 24380543011087797121, 981149507717921468737, 41653436572936172408455, 1860174362332664149119607

Offset: 0

Paul Barry, Aug 29 2005

In general, e.g.f. exp(x(1+ax)/(1-bx)) has general term sum{i=0..n, sum{j=0..n, a^j*b^(n-i-j)*C(i,j)C(n-j-1,n-i-j)*n!/i!}}.

Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is a purely periodic sequence whose period divides k. For example, taken modulo 14 the sequence becomes [1, 1, 7, 13, 3, 7, 5, 1, 1, 7, 13, 3, 7, 5, ...], a periodic sequence of period 7. Cf. A047974. - Peter Bala, Jul 15 2022

With[{nn=20},CoefficientList[Series[Exp[(x(x+1))/(1-2x)],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Sep 21 2011 *)

E.g.f.: exp(x*(1+x)/(1-2*x)).
a(n) = Sum_{i = 0..n} Sum_{j = 0..n} 2^(n-i-j)*C(i, j)*C(n-j-1, n-i-j)*n!/i!.
a(n) ~ 3^(1/4) * 2^(n-1) * n^(n-1/4) * exp(sqrt(3*n)-n-5/8). - Vaclav Kotesovec, Sep 25 2013
Conjecture: a(n) +(-4*n+3)*a(n-1) +2*(n-1)*(2*n-5)*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Feb 20 2015
The e.g.f. A(x) satisfies the differential equation (4*x^2 - 4*x + 1)*A'(x) + (2*x^2 - 2*x - 1)*A(x) = 0 with A(0) = 1. Mathar's conjectured recurrence follows easily from this. - Peter Bala, Jul 15 2022

A382139 Number of matchings of [2n] with no coupled arcs.

Original entry on oeis.org

1, 1, 1, 9, 81, 705, 7665, 100905, 1524705, 26022465, 496042785, 10445342985, 240779831985, 6030718158465, 163087008669585, 4735950860666025, 146987669673669825, 4855606200012593025, 170101350767940617025, 6298861062893921346825, 245834199405298416337425

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, eq (17) at r=1, s=2.

Cf. A047974, A053871, A382134, A067994, A001147.

g:= exp(-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x^2)/sqrt(1-2*x).
Exponential convolution of A067994 and A001147.
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) +2*(n-1)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0.

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A064571 Binomial transform of (3n)!.

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A112243 Expansion of exp(x(1+x)/(1-2x)).

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A382139 Number of matchings of [2n] with no coupled arcs.

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cp's OEIS Frontend

A064571 Binomial transform of (3n)!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A112243 Expansion of exp(x*(1+x)/(1-2*x)).

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

A382139 Number of matchings of [2n] with no coupled arcs.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A112243 Expansion of exp(x(1+x)/(1-2x)).