A236471 Riordan array ((1-x)/(1-2*x), x(1-x)/(1-2*x)^2).

1, 1, 1, 2, 4, 1, 4, 13, 7, 1, 8, 38, 33, 10, 1, 16, 104, 129, 62, 13, 1, 32, 272, 450, 304, 100, 16, 1, 64, 688, 1452, 1289, 590, 147, 19, 1, 128, 1696, 4424, 4942, 2945, 1014, 203, 22, 1, 256, 4096, 12896, 17584, 13073, 5823, 1603, 268, 25, 1, 512, 9728

Offset

0,4

Comments

Row sums are A052936(n).

Diagonal sums are A121449(n).

The triangle T'(n,k) = T(n,k)*(-1)^(n+k) is the inverse of the Riordan array in A090285.

Links

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Formula

T(n,0) = A011782(n), T(n,1) = A049611(n), T(n,n) = A000012(n) = 1, T(n+1,n) = A016777(n), T(n+2,n) = A062708(n+1).

G.f.: (2*x^2-3*x+1)/((x^2-x)*y+4*x^2-4*x+1). - Vladimir Kruchinin, Apr 21 2015

T(n,k) = Sum_{m=0..n} C(m+k,2*k)*C(n-1,n-m). - Vladimir Kruchinin, Apr 21 2015

Example

Triangle begins:
1;
1, 1;
2, 4, 1;
4, 13, 7, 1;
8, 38, 33, 10, 1;
16, 104, 129, 62, 13, 1;
32, 272, 450, 304, 100, 16, 1;
64, 688, 1452, 1289, 590, 147, 19, 1;

Mathematica

CoefficientList[CoefficientList[Series[(2*x^2-3*x+1)/((x^2-x)*y +4*x^2 - 4*x+1), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten (* G. C. Greubel, Apr 19 2018 *)

Prog

(Maxima)
T(n, k):=sum(binomial(m+k, 2*k)*binomial(n-1, n-m), m, 0, n); /* Vladimir Kruchinin, Apr 21 2015 */
(PARI) for(n=0, 20, for(k=0, n, print1(sum(m=0, n, binomial(m+k, 2*k)* binomial(n-1, n-m)), ", "))) \\ G. C. Greubel, Apr 19 2018

Crossrefs

Cf. A011782, A049611, A000012, A016777, A062708.

Sequence in context: A085111 A181332 A204021 * A220328 A220935 A221290

Adjacent sequences: A236468 A236469 A236470 * A236472 A236473 A236474

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 26 2014

Status

approved