A251688
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1+2*x)^n*(1+3*x)^n.
Original entry on oeis.org
1, 1, 25, 61, 336, 1200, 3600, 13500, 32400, 118800, 259200, 939600, 1944000, 6998400, 13996800, 50155200, 97977600, 349920000, 671846400, 2393452800, 4534963200, 16124313600, 30233088000, 107327462400, 199538380800, 707454259200, 1306069401600, 4625662464000, 8489451110400
Offset: 0
G.f.: A(x) = 1 + x + 25*x^2 + 61*x^3 + 336*x^4 + 1200*x^5 + 3600*x^6 +...
where
log(A(x)) = (1 + 5^2*x + 6^2*x^2)/A(x) * x +
(1 + 10^2*x + 37^2*x^2 + 60^2*x^3 + 36^2*x^4)/A(x)^2 * x^2/2 +
(1 + 15^2*x + 93^2*x^2 + 305^2*x^3 + 558^2*x^4 + 540^2*x^5 + 216^2*x^6)/A(x)^3 * x^3/3 +
(1 + 20^2*x + 174^2*x^2 + 860^2*x^3 + 2641^2*x^4 + 5160^2*x^5 + 6264^2*x^6 + 4320^2*x^7 + 1296^2*x^8)/A(x)^4 * x^4/4 +
(1 + 25^2*x + 280^2*x^2 + 1850^2*x^3 + 7985^2*x^4 + 23525^2*x^5 + 47910^2*x^6 + 66600^2*x^7 + 60480^2*x^8 + 32400^2*x^9 + 7776^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + 5*x + 6*x^2)^n.
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{a(n)=polcoeff( (1+x)*(1+4*x^2)*(1+9*x^2)*(1+36*x^3) / ((1-6*x^2)^2 +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))
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{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff(((1+2*x)*(1+3*x) +x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
A248876
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..3*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ) where T(n,k) equals the coefficient of x^k in (1+x+x^2+x^3)^n.
Original entry on oeis.org
1, 1, 1, 2, 4, 8, 13, 24, 45, 85, 161, 305, 582, 1116, 2149, 4152, 8049, 15653, 30528, 59695, 117012, 229880, 452565, 892703, 1764099, 3492029, 6923494, 13747483, 27335873, 54427621, 108505081, 216568556, 432740907, 865610375, 1733227339, 3473805680, 6968708734, 13991916510, 28116598325
Offset: 0
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 13*x^6 + 24*x^7 +...
where
log(A(x)) = (1 + x + x^2 + x^3)/A(x) * x +
(1 + 2^2*x + 3^2*x^2 + 4^2*x^3 + 3^2*x^4 + 2^2*x^5 + x^6)/A(x)^2 * x^2/2 +
(1 + 3^2*x + 6^2*x^2 + 10^2*x^3 + 12^2*x^4 + 12^2*x^5 + 10^2*x^6 + 6^2*x^7 + 3^2*x^8 + x^9)/A(x)^3 * x^3/3 +
(1 + 4^2*x + 10^2*x^2 + 20^2*x^3 + 31^2*x^4 + 40^2*x^5 + 44^2*x^6 + 40^2*x^7 + 31^2*x^8 + 20^2*x^9 + 10^2*x^10 + 4^2*x^11 + x^12)/A(x)^4 * x^4/4 +
which involves the squares of the coefficients in (1 + x + x^2 + x^3)^n.
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/* By Definition: */
{T(n,k)=polcoeff((1 + x + x^2 + x^3 + x*O(x^k))^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, min(3*m,n-m), T(m,k)^2 * x^k) / (A +x*O(x^n))^m * x^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
A255839
G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..3*n} binomial(3*n,k)^2 * x^k] / A(x)^n * x^n/n ).
Original entry on oeis.org
1, 1, 9, 18, 64, 172, 477, 1368, 3681, 10485, 28701, 80829, 225090, 632160, 1778553, 5010948, 14181849, 40161357, 114151716, 324873027, 926918784, 2649218580, 7585705665, 21758756931, 62508649059, 179859399129, 518234494662, 1495275239115, 4319808231645, 12495043092609, 36183457564425
Offset: 0
G.f.: A(x) = 1 + x + 9*x^2 + 18*x^3 + 64*x^4 + 172*x^5 + 477*x^6 +...
where
log(A(x)) = (1 + 3^2*x + 3^2*x^2 + x^3)/A(x) * x +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)/A(x)^2 * x^/2 +
(1 + 9^2*x + 36^2*x^2 + 84^2*x^3 + 126^2*x^4 + 126^2*x^5 + 84^2*x^6 + 36^2*x^7 + 9^2*x^8 + x^9)/A(x)^3 * x^3/3 +
(1 + 12^2*x + 66^2*x^2 + 220^2*x^3 + 495^2*x^4 + 792^2*x^5 + 924^2*x^6 + 792^2*x^7 + 495^2*x^8 + 220^2*x^9 + 66^2*x^10 + 12^2*x^11 + x^12)/A(x)^4 * x^4/4 +...
which involves the squares of the coefficients in (1 + x)^(3*n).
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/* By Definition: */
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, min(3*m,n-m), binomial(3*m,k)^2 * x^k) / (A +x*O(x^n))^m * x^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
Showing 1-3 of 3 results.
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