A344502 a(n) = Sum_{k=0..n} binomial(n, k)^2 * hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).

1, 2, 7, 29, 128, 587, 2759, 13190, 63844, 311948, 1535488, 7602971, 37829455, 188989166, 947399951, 4763280965, 24009574400, 121291129748, 613939110308, 3112989719080, 15809048927000, 80397234851080, 409378690617344, 2086928493438299, 10649867701045871

Offset

0,2

Comments

Binomial convolution of the Motzkin numbers.

Links

Table of n, a(n) for n=0..24.

Formula

a(n) ~ sqrt((76 + 5*(38*(247 - 27*sqrt(57)))^(1/3) + 5*(38*(247 + 27*sqrt(57)))^(1/3))/57)/4 * ((1261 + 57*sqrt(57))^(1/3)/6 + 56/(3*(1261 + 57*sqrt(57))^(1/3)) + 5/3)^n / sqrt(Pi*n). - Vaclav Kotesovec, May 24 2021

Conjecture D-finite with recurrence -4*(3035*n-11997) *(2*n+1) *(n+1) *a(n) +2*(153546*n^3-490325*n^2-62942*n+71982) *a(n-1) +2*(-452090*n^3+1745622*n^2-1405285*n-226200) *a(n-2) -2 *(n-2)*(83741*n^2-200458*n-19650) *a(n-3) -3*(n-2) *(n-3) *(46423*n+6938) *a(n-4)=0. - R. J. Mathar, Nov 02 2021

Maple

a := n -> add(binomial(n, k)^2 * hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n); seq(simplify(a(n)), n = 0..24);

Crossrefs

Cf. A064189 (Motzkin numbers), A344503.

Cf. A348840.

Sequence in context: A126568 A150663 A054321 * A150664 A193040 A200755

Adjacent sequences: A344499 A344500 A344501 * A344503 A344504 A344505

Keyword

nonn

Author

Peter Luschny, May 23 2021

Status

approved