A005554 -id:A005554 - cp's OEIS frontend

A034929 A triangle of Motzkin ballot numbers, read by rows.

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 6, 6, 4, 1, 5, 10, 13, 13, 9, 1, 6, 15, 24, 30, 30, 21, 1, 7, 21, 40, 59, 72, 72, 51, 1, 8, 28, 62, 105, 148, 178, 178, 127, 1, 9, 36, 91, 174, 276, 378, 450, 450, 323, 1, 10, 45, 128, 273, 480, 730, 980, 1158, 1158, 835, 1, 11, 55, 174, 410, 791

Offset: 1

N. J. A. Sloane

Mirror image of A091836. Row sums are the Motzkin numbers (A001006). T(n,n-1)=A001006(n-2) (the Motzkin numbers). T(n,n-2)=A005554(n-1).

			Triangle begins:
[1],
[1, 1],
[1, 2, 1],
[1, 3, 3, 2],
[1, 4, 6, 6, 4],
[1, 5, 10, 13, 13, 9],
[1, 6, 15, 24, 30, 30, 21],
[1, 7, 21, 40, 59, 72, 72, 51]

M. Aigner, Motzkin numbers, Europ. J. Comb. 19 (1998), 663-675.

Cf. A001006, A091836, A005554.

G.f.= 2(1+tz)/[1-2z+tz-2tz^2+sqrt(1-2tz-3t^2*z^2)].

Edited by Emeric Deutsch, Mar 11 2004

A228338 Third diagonal of Catalan difference table (A059346).

Original entry on oeis.org

5, 9, 19, 43, 102, 250, 628, 1608, 4181, 11009, 29295, 78655, 212815, 579675, 1588245, 4374285, 12103407, 33628827, 93786969, 262450881, 736710360, 2073834420, 5853011850, 16558618510, 46949351275, 133390812255, 379708642289, 1082797114049, 3092894319078, 8848275403642

Offset: 0

N. J. A. Sloane, Aug 29 2013

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A059346, A001006, A005043, A005554.

a := n -> 5*(-1)^n*hypergeom([7/2, -n], [5], 4):
seq(simplify(a(n)), n=0..29); # Peter Luschny, May 25 2021

CoefficientList[Series[-(x+1)^(5/2)*Sqrt[1-3*x]/(2*x^4)-1/2*(- 1 - x + 3*x^2 + 7*x^3)/x^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 14 2014 *)

x='x+O('x^50); Vec(-(x+1)^(5/2)*sqrt(1-3*x)/(2*x^4)-1/2*(-1-x+3*x^2+7*x^3)/x^4) \\ G. C. Greubel, May 31 2017

From Vaclav Kotesovec, Feb 14 2014: (Start)
Recurrence: (n+4)*a(n) = (2*n+7)*a(n-1) + 3*(n-1)*a(n-2).
G.f.: -(x+1)^(5/2)*sqrt(1-3*x)/(2*x^4)-1/2*(-1-x+3*x^2+7*x^3)/x^4.
a(n) ~ 8 * 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). (End)
a(n) = 5*(-1)^n*hypergeom([7/2, -n], [5], 4). - Peter Luschny, May 25 2021

Terms a(21) onward added by G. C. Greubel, May 31 2017

A378858 G.f. A(x) satisfies A(x) = ( 1 + x/(1 - x*A(x)^(3/4)) )^4.

Original entry on oeis.org

1, 4, 10, 32, 119, 468, 1934, 8256, 36135, 161276, 731158, 3357748, 15587004, 73021200, 344786056, 1639145180, 7839483967, 37692820908, 182087119582, 883358016328, 4301799946048, 21021519618724, 103049029114618, 506608410994868, 2497162797380145, 12338908560964968

Offset: 0

Seiichi Manyama, Dec 09 2024

nonn

Cf. A364742, A365119, A378730.

Cf. A005554, A378801.

a(n, r=4, s=1, t=0, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: A(x) = (1 + x*B(x))^4 where B(x) is the g.f. of A364742.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

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A034929 A triangle of Motzkin ballot numbers, read by rows.

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A228338 Third diagonal of Catalan difference table (A059346).

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A378858 G.f. A(x) satisfies A(x) = ( 1 + x/(1 - x*A(x)^(3/4)) )^4.

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