A184554 Self-convolution equals A184553.

1, 3, 35, 474, 6891, 104360, 1623050, 25718472, 413215707, 6710439939, 109904635992, 1812533851286, 30064278051066, 501094410251724, 8386624585529736, 140867399832201392, 2373517896651329211

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Formula

Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(3n+k,n-k)*C(4n-k,k).

From Vaclav Kotesovec, Oct 05 2020: (Start)

Recurrence: 214990848*(n-1)*n*(2*n - 1)*(3*n - 2)*(3*n - 1)*(6*n - 5)*(6*n - 1)*(186273819397795248*n^10 - 3808534256911136592*n^9 + 34796629832777934044*n^8 - 187043184670288993620*n^7 + 654924444499586105253*n^6 - 1560497773606771079631*n^5 + 2561855600168977896561*n^4 - 2860703663001433319865*n^3 + 2078954085287299115314*n^2 - 887684422175942220312*n + 169072062403455034560)*a(n) = 4608*(n-1)*(198724237800527802240275328*n^16 - 4858079103240534545068477440*n^15 + 54642880050599236064003042400*n^14 - 374971611530329495568194400064*n^13 + 1755012126559348835693811861336*n^12 - 5932874750520572573109360538464*n^11 + 14963200004531199654932033618980*n^10 - 28673115580838375915179786787512*n^9 + 42110417881849893372861579821049*n^8 - 47452926131517602493132103229892*n^7 + 40787440281095755641655153245870*n^6 - 26377403019510743431336185730824*n^5 + 12533524812443184688591954943537*n^4 - 4209478808341707391000183637804*n^3 + 936469297986509962567040719500*n^2 - 122249305263206439707648569200*n + 6971356419682529260674288000)*a(n-1) - 56*(144127472482539322780646079360*n^17 - 3883898817531102082800910713408*n^16 + 48508159847016350153460641621824*n^15 - 372595836943650102686279598364576*n^14 + 1969341905770556875691823842890528*n^13 - 7592461765467638827554889057000528*n^12 + 22081067675956591873212706692285658*n^11 - 49409399253505762430059464298609003*n^10 + 85975527766708349125221818435201054*n^9 - 116774106223860017685029870069674610*n^8 + 123519332779512059055811665446677502*n^7 - 100900018919730898675966601457772931*n^6 + 62669067428714316682185812483083058*n^5 - 28859522902379638250482670957736704*n^4 + 9468958420673660795040733200084216*n^3 - 2072765338981605418905542909841840*n^2 + 268050646552599842334757631726400*n - 15244779045642670870731418176000)*a(n-2) - 5764801*(n-2)*(7*n - 20)*(7*n - 19)*(7*n - 18)*(7*n - 17)*(7*n - 16)*(7*n - 15)*(186273819397795248*n^10 - 1945796062933184112*n^9 + 8902143393478490876*n^8 - 23424520929131008820*n^7 + 39128411618346831497*n^6 - 43181027932317815765*n^5 + 31728210165988475069*n^4 - 15234320478860939075*n^3 + 4539474588760473630*n^2 - 750365114092889508*n + 51518300147130960)*a(n-3).

a(n) ~ 7^(7*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(6*n + 2) * 3^(6*n + 1/4)).

(End)

Example

G.f.: A(x) = 1 + 3*x + 35*x^2 + 474*x^3 + 6891*x^4 + 104360*x^5 +...
A(x)^2 = 1 + 6*x + 79*x^2 + 1158*x^3 + 17851*x^4 + 283246*x^5 + 4579306*x^6 +...+ A184553(n)*x^n +...

Prog

(PARI) {a(n)=local(A2=sum(m=0, n, sum(k=0, m, binomial(3*m+k, m-k)*binomial(4*m-k, k))*x^m+x*O(x^n))); polcoeff(A2^(1/2), n)}

Crossrefs

Cf. A184553, A183161.

Sequence in context: A179135 A100033 A259557 * A046032 A304191 A009071

Adjacent sequences: A184551 A184552 A184553 * A184555 A184556 A184557

Keyword

nonn

Author

Paul D. Hanna, Jan 16 2011

Status

approved