OFFSET
0,4
LINKS
Manuel Kauers and Christoph Koutschan, Table of n, a(n) for n = 0..836 (terms 0..40 from Alois P. Heinz).
M. Kauers and C. Koutschan, Some D-finite and some possibly D-finite sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.
FORMULA
From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)
a(n) = 2*Sum_{j=1..n} (Sum_{i=1..n} N'(i, j) + (n-j-1)*N'(j, n)) - 2*binomial(2*n, n) + N'(n, n) + 3 for n>0, where N'(n, k) = (binomial(n+k-1, n-1)*binomial(n+k-1, n))/(n+k-1) denotes the Narayana number N(n+k-1, k).
Recurrence: -2*(n+3)*(n+4)^2*(2*n+7)*(118125*n^10 + 1308375*n^9 + 6016950*n^8 + 14827410*n^7 + 20875365*n^6 + 15986367*n^5 + 4449768*n^4 - 2342808*n^3 - 2279152*n^2 - 660240*n - 64000)*a(n+4) + (n+3)*(10040625*n^13 + 210555000*n^12 + 1942194375*n^11 + 10361592450*n^10 + 35325144315*n^9 + 80085358620*n^8 + 121180651809*n^7 + 117919810482*n^6 + 64349576684*n^5 + 7017979960*n^4 - 14571577344*n^3 - 9566235392*n^2 - 2428639744*n - 221347840)*a(n+3) + (-42170625*n^14 - 981642375*n^13 - 10209053025*n^12 - 62559627795*n^11 - 250621464735*n^10 - 687475711989*n^9 - 1311094658043*n^8 - 1718884004625*n^7 - 1471227292164*n^6 - 691541238960*n^5 - 14462120192*n^4 + 188403075920*n^3 + 108128100864*n^2 + 25779317504*n + 2257059840)*a(n+2) + 2*(2*n+3)*(10040625*n^13 + 215870625*n^12 + 2048259375*n^11 + 11279217825*n^10 + 39828085965*n^9 + 93825035775*n^8 + 147951032109*n^7 + 150478534491*n^6 + 86482913102*n^5 + 11547320420*n^4 - 18788310824*n^3 - 12713618176*n^2 - 3178474112*n - 272670720)*a(n+1) - 8*n*(2*n-1)*(2*n+1)*(2*n+3)*(118125*n^10 + 2489625*n^9 + 23107950*n^8 + 124239510*n^7 + 427851585*n^6 + 984186117*n^5 + 1527319428*n^4 + 1572814284*n^3 + 1022652512*n^2 + 375620224*n + 58236160)*a(n) = 0. (End)
a(n) ~ 25 * 2^(4*n - 3) / (9*Pi*n^2) . - Vaclav Kotesovec, Mar 08 2023
MATHEMATICA
Nara[i_, j_] := 1/(i+j-1)*Binomial[i+j-1, i]*Binomial[i+j-1, i-1];
Prepend[Table[2*Sum[Sum[Nara[i, j], {i, n}] + (n-j-1)*Nara[j, n], {j, n}] - 2*Binomial[2*n, n] + Nara[n, n] + 3, {n, 100}], 1] (* Manuel Kauers and Christoph Koutschan, Mar 02 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 07 2019
STATUS
approved