A118933 -id:A118933 - cp's OEIS frontend

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

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).

A118934 E.g.f.: exp(x + x^4/4).

Original entry on oeis.org

1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, 57961, 209881, 1874071, 17842111, 117303187, 575683291, 5691897121, 65641390081, 544238393041, 3362783785777, 36455473647271, 485442581801311, 4828464958268491, 35900587138847971, 423276450114749617, 6318491163509870401

Offset: 0

Paul D. Hanna, May 06 2006

nonn

Equals row sums of triangle A118933.

These are the telephone numbers T^(4)n of [Artioli et al., p. 7]. - _Eric M. Schmidt, Oct 12 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Vaclav Kotesovec, Graph - asymptotic (20000 terms)

Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), this sequence (m=4), A052501 (m=5), A293588 (m=6), A053497 (m=7).

Cf. A118933.

F:=Factorial; [(&+[F(n)/(4^j*F(j)*F(n-4*j)): j in [0..Floor(n/4)]]): n in [0..30]]; // G. C. Greubel, Mar 07 2021

With[{nn=30},CoefficientList[Series[Exp[x+x^4/4],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 26 2013 *)
Table[Sum[n!/(4^k*k!*(n-4*k)!), {k,0,n/4}], {n,0,30}]

a(n)=if(n<0,0,if(n==0,1,a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4)))

f=factorial; [sum(f(n)/(4^j*f(j)*f(n-4*j)) for j in (0..n/4)) for n in (0..30)] # G. C. Greubel, Mar 07 2021

a(n) = a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4) for n>=4, with a(0)=a(1)=a(2)=a(3)=1.
a(n) ~ 1/2 * n^(3*n/4) * exp(n^(1/4)-3*n/4). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/4)} n!/(4^k*k!*(n-4*k)!). - G. C. Greubel, Mar 07 2021

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

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

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A118934 E.g.f.: exp(x + x^4/4).

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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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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A118934 E.g.f.: exp(x + x^4/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

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