A046527 -id:A046527 - cp's OEIS frontend

A046814 Row sums of triangle A046527.

1, 2, 8, 37, 179, 881, 4369, 21746, 108444, 541362, 2704158, 13512392, 67534828, 337584992, 1687627800, 8437136085, 42182258715, 210899507685, 1054456597965, 5272139698215, 26360193558735, 131799177579015

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A046527, A046714, A046748.

[n le 1 select 1 else 5*Self(n-1) - 3*Catalan(n-1)/(2*n-3): n in [1..40]]; // G. C. Greubel, Jul 28 2024

CoefficientList[Series[(1-4*x)*(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x,0,40}], x] (* G. C. Greubel, Jul 28 2024 *)

@CachedFunction
def A046814(n): return 1 if n==0 else 5*A046814(n-1) - 3*catalan_number(n)/(2*n-1)
[A046814(n) for n in range(41)] # G. C. Greubel, Jul 28 2024

G.f.: c(x) * (1-4*x) / (1-5*x), where c(x) = g.f. for Catalan A000108.
a(n) = C(n) + A046714(n-1) with A046714(-1) = 0 and C(n) = A000108(n) are the Catalan numbers.
a(n) = C(n) + (5^n - A046748(n))/2.
a(n) = 5*a(n-1) - 3*C(n)/(2*n-1), a(0)=1.
D-finite with recurrence a(n) = (9*n-1)*a(n-1)/(n+1) - 10*(2*n-3)*a(n-2)/(n+1), n >= 2, a(0)=1, a(1)=2.

Offset corrected by Sean A. Irvine, Apr 25 2021

A048870 Triangle of coefficients of certain Sheffer-polynomials.

Original entry on oeis.org

1, 1, 1, 4, 10, 1, 30, 132, 27, 1, 336, 2232, 696, 52, 1, 5040, 46320, 19500, 2200, 85, 1, 95040, 1141920, 606960, 91800, 5340, 126, 1, 2162160, 32639040, 20991600, 3986640, 310170, 11004, 175, 1, 57657600, 1061746560, 802287360, 183550080

Offset: 0

Wolfdieter Lang

s(n,x) := sum(a(n,m)*x^m,m=0..n) are monic polynomials satisfying s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(A048786(n,m)*x^m, m=1..n) (row polynomials of triangle A048786) and p(0,x)=1. In the umbral calculus (see reference) the s(n,x) are called Sheffer polynomials for(c(t/(1+4*t)),t/(1+4*t)), where c(x) = g.f. for Catalan numbers A000108. a(n,0) = A001761(n-2) = n!*A000108(n).

S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

A046527, A048786, A000108, A001761.

a(n, m) = (n!/m!)*A046527(n, m) = (n!/m!)*binomial(n, m-1)*(4^(n-m+1)-binomial(2*n, n)/binomial(2*(m-1), m-1))/2, n >= m >= 0, a(n, m) := 0, n

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cp's OEIS Frontend

A046814 Row sums of triangle A046527.

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