A118932 -id:A118932 - cp's OEIS frontend

A118930 E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).

1, 1, 2, 4, 13, 41, 166, 652, 3494, 18118, 114076, 681176, 5016892, 35377564, 288204008, 2232198256, 21124254181, 191779964597, 2011347229114, 19840403629108, 231266808172181, 2553719667653281, 31743603728993542

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals invariant column vector V that satisfies matrix product A100861*V = V, where Bessel numbers A100861(n,k) = n!/[k!(n-2k)!*2^k] give the number of k-matchings of the complete graph K(n).

Equals Lim_{n->inf.} A144299^n, if A144299 is considered an infinite lower triangular matrix. - Gary W. Adamson, Dec 08 2008

			E.g.f. A(x) = exp( x + x^2/2 + x^4/2^3 + x^8/2^7 + x^16/2^15 +...)
= 1 + 1*x + 2*x^2/2! + 4*x^3/3! + 13*x^4/4! + 41*x^5/5!+ 166*x^6/6!+...
Using coefficients A100861(n,k) = n!/[k!(n-2k)!*2^k]:
a(5) = 1*a(0) +10*a(1) +15*a(2) = 1*1 +10*1 +15*2 = 41.
a(6) = 1*a(0) +15*a(1) +45*a(2) +15*a(3) = 1*1 +15*1 +45*2 +15*4 = 166.

Cf. A100861; variants: A118932, A118935.
Equals row sums of triangle A152685. - Gary W. Adamson, Dec 10 2008
Cf. A144299. - Gary W. Adamson, Dec 08 2008

A118930 := proc(n)
    option remember;
    if n<= 1 then
        1 ;
    else
        n!*add(procname(k)/k!/(n-2*k)!/2^k,k=0..n/2) ;
    end if;
end proc;
seq(A118930(n),n=0..10) ; # R. J. Mathar, Aug 19 2014

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, 2k] (2k-1)!! a[k], {k, 0, n/2}]];
a /@ Range[0, 22] (* Jean-François Alcover, Mar 26 2020 *)

{a(n)=if(n==0,1,sum(k=0,n\2,n!/(k!*(n-2*k)!*2^k)*a(k)))}
for(n=0,30,print1(a(n),", "))

/* Defined by E.G.F.: */
{a(n)=n!*polcoeff( exp(sum(k=0,#binary(n),x^(2^k)/2^(2^k-1))+x*O(x^n)),n,x)}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..[n/2]} n!/[k!*(n-2*k)!*2^k] * a(k), with a(0)=1. a(n) = Sum_{k=0..[n/2]} A100861(n,k)*a(k), with a(0)=1.

A118931 Triangle, read by rows, where T(n,k) = n!/(k!(n-3k)!3^k) for n>=3k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 8, 1, 20, 1, 40, 40, 1, 70, 280, 1, 112, 1120, 1, 168, 3360, 2240, 1, 240, 8400, 22400, 1, 330, 18480, 123200, 1, 440, 36960, 492800, 246400, 1, 572, 68640, 1601600, 3203200, 1, 728, 120120, 4484480, 22422400, 1, 910, 200200, 11211200, 112112000, 44844800

Offset: 0

Paul D. Hanna, May 06 2006

Row n contains 1+floor(n/3) terms. Row sums yield A001470. Given column vector V = A118932, then V is invariant under matrix product T*V = V, or, A118932(n) = Sum_{k=0..n} T(n,k)*A118932(k). Given C = Pascal's triangle and T = this triangle, then matrix product M = C^-1*T yields M(3n,n) = (3*n)!/(n!*3^n), 0 otherwise (cf. A100861 formula due to Paul Barry).

			Triangle T begins:
  1;
  1;
  1;
  1,   2;
  1,   8;
  1,  20;
  1,  40,    40;
  1,  70,   280;
  1, 112,  1120;
  1, 168,  3360,   2240;
  1, 240,  8400,  22400;
  1, 330, 18480, 123200;
  1, 440, 36960, 492800, 246400;

G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Cf. A001470 (row sums), A118932 (invariant vector).

Variants: A100861, A118933.

F:= Factorial;
[n lt 3*k select 0 else F(n)/(3^k*F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

Trow := n -> seq(n!/(j!*(n - 3*j)!*(3^j)), j = 0..n/3):
seq(Trow(n), n = 0..14); # Peter Luschny, Jun 06 2021

T[n_,k_]:= If[n<3*k, 0, n!/(3^k*k!*(n-3*k)!)];
Table[T[n,k], {n,0,20}, {k,0,Floor[n/3]}]//Flatten (* G. C. Greubel, Mar 07 2021 *)

T(n,k)=if(n<3*k,0,n!/(k!*(n-3*k)!*3^k))

f=factorial;
flatten([[0 if n<3*k else f(n)/(3^k*f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

E.g.f.: A(x,y) = exp(x + y*x^3/3).

A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 454105, 4309921, 23452111, 592433161, 6789801877, 46254009985, 893881991731, 11548704851041, 93501748795441, 4828847934591937, 83867376656907415, 823025819684123641, 33409213329178701421, 640457721676922946721

Offset: 0

Paul D. Hanna, May 07 2006

nonn

E.g.f. of triangle A118394 is: exp(x+y*x^3), where A118394(n,k) = n!/k!/(n-3*k)!. More generally, given a triangle with e.g.f.: exp(x+y*x^b), the eigenvector will have e.g.f.: exp( Sum_{n>=0} x^(b^n) ).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A118394, A118395; variants: A118393, A118932.

a:= proc(n) option remember; `if`(n=0, 1, add((j-> j!*
      a(n-j)*binomial(n-1, j-1))(3^i), i=0..ilog[3](n)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 01 2017

a[n_] := a[n] = If[n==0, 1, Sum[Function[j, j! a[n-j] Binomial[n-1, j-1]][3^i], {i, 0, Log[3, n]}]];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

{a(n)=n!*polcoeff(exp(sum(k=0,ceil(log(n+1)/log(3)),x^(3^k))+x*O(x^n)),n)}

a(n) = Sum_{k=0..[n/3]} n!/k!/(n-3*k)! *a(k) for n>=0, with a(0)=1.

A118935 E.g.f.: A(x) = exp( Sum_{n>=0} x^(4^n)/4^((4^n-1)/3) ).

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, 57961, 209881, 1874071, 17842111, 117303187, 575683291, 26124309121, 412992394081, 3670397429041, 23161791013777, 729420726627271, 13374596287229311, 143560108604864491

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals invariant column vector V that satisfies matrix product A118933*V = V, where A118933(n,k) = n!/[k!(n-4k)!*4^k] for n>=4*k>=0; thus a(n) = Sum_{k=0..[n/4]} A118933(n,k)*a(k), with a(0)=1.

			E.g.f. A(x) = exp( x + x^4/4 + x^16/4^5 + x^64/3^21 + x^256/3^85 +..)
= 1 + 1*x + 1*x^2/2! + 1*x^3/3! + 7*x^4/4! + 31*x^5/5!+ 91*x^6/6!+...

Cf. A118933; variants: A118930, A118932.

a(n)=if(n==0,1,sum(k=0,n\4,n!/(k!*(n-4*k)!*4^k)*a(k)))

/* Defined by E.G.F.: */ a(n)=n!*polcoeff( exp(sum(k=0,ceil(log(n+1)/log(4)),x^(4^k)/4^((4^k-1)/3))+x*O(x^n)),n,x)

a(n) = Sum_{k=0..[n/4]} n!/[k!*(n-4*k)!*4^k] * a(k), with a(0)=1.

cp's OEIS Frontend

A118930 E.g.f.: A(x) = exp( Sum_{n>=0} x^(2^n)/2^(2^n-1) ).

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A118931 Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0.

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A118396 Eigenvector of triangle A118394; E.g.f.: exp( Sum_{n>=0} x^(3^n) ).

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A118935 E.g.f.: A(x) = exp( Sum_{n>=0} x^(4^n)/4^((4^n-1)/3) ).

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PARI

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A118931 Triangle, read by rows, where T(n,k) = n!/(k!(n-3k)!3^k) for n>=3k>=0.