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a(n) ~ 25 * 2^(4*n - 3) / (9*Pi*n^2) . - Vaclav Kotesovec, Mar 08 2023

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M. Kauers and C. Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some possibly D-finite sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023.

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Manuel Kauers and Christoph Koutschan, <a href="/A306322/b306322_2.txt">Table of n, a(n) for n = 0..836</a> (terms 0..40 from Alois P. Heinz).

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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).

Manuel Kauers and Christoph Koutschan, <a href="/A306322/b306322_2.txt">Table of n, a(n) for n = 0..836</a> (terms 0..40 from Alois P. Heinz).

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, 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*binomialn+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'(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, where N' + 108128100864*n^2 + 25779317504*n + 2257059840)*a(n+2) + 2*(2*n, k+3) = *(binomial10040625*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+k1) - 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)

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)