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 A181069 Expansion of l.g.f.: Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n. 3
 1, 3, 28, 275, 3126, 37632, 475056, 6192531, 82754650, 1127504378, 15603575208, 218727171104, 3099183987004, 44315462038200, 638663235342528, 9267264584278419, 135279095477748642, 1985221072388231742 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS G. C. Greubel, Table of n, a(n) for n = 1..500 FORMULA a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * binomial(n,k). Recurrence: (n-1)^2*n^3*(10*n^2 - 25*n + 16)*a(n) = 2*(n-1)^2*(60*n^5 - 240*n^4 + 341*n^3 - 225*n^2 + 90*n - 16)*a(n-1) + 4*(n-2)^2*n*(4*n - 7)*(4*n - 5)*(10*n^2 - 5*n + 1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014 a(n) ~ 2^(4*n-5/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Mar 06 2014 EXAMPLE L.g.f.: L(x) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 +... which equals the series: L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x + (1 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 + ...)*x^2/2 + (1 + 3^4*x + 6^4*x^2 + 10^4*x^3 + 15^4*x^4 + 21^4*x^5 + ...)*x^3/3 + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 + ...)*x^4/4 + (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 + 70^4*x^4 + 126^4*x^5 + ...)*x^5/5 + (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 + ...)*x^6/6 + (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 + ...)*x^7/7 + ... Exponentiation yields the g.f. of A181068: exp(L(x)) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +... MAPLE A181069:= n-> add( binomial(n, k)*binomial(n-1, k)^3, k=0..n-1); seq(A181069(n), n=1..20); # G. C. Greubel, Apr 05 2021 MATHEMATICA Table[Sum[Binomial[n-1, k]^3 * Binomial[n, k], {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 06 2014 *) PROG (PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^4*n/(n-k))} (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^4*x^k)*x^m/m)+x*O(x^n), n)} for(n=1, 20, print1(a(n), ", ")) (Magma) [(&+[Binomial(n, k)*Binomial(n-1, k)^3: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Apr 05 2021 (Sage) [sum( binomial(n, k)*binomial(n-1, k)^3 for k in (0..n-1) ) for n in (1..20)] # G. C. Greubel, Apr 05 2021 CROSSREFS Cf. A181067 (variant), A181068. Sequence in context: A076723 A198887 A026114 * A239297 A287884 A250890 Adjacent sequences: A181066 A181067 A181068 * A181070 A181071 A181072 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 08 2010 STATUS approved

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Last modified September 22 10:13 EDT 2023. Contains 365520 sequences. (Running on oeis4.)