cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A251687 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 + x + 2*x^2)^n.

Original entry on oeis.org

1, 1, 1, 5, 8, 8, 16, 28, 48, 80, 128, 208, 320, 512, 768, 1216, 1792, 2816, 4096, 6400, 9216, 14336, 20480, 31744, 45056, 69632, 98304, 151552, 212992, 327680, 458752, 704512, 983040, 1507328, 2097152, 3211264, 4456448, 6815744, 9437184, 14417920, 19922944, 30408704, 41943040
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2015

Keywords

Comments

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x + r*x^2)^n, then G(x) = (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / (1-p*r*x^2)^2.

Examples

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 28*x^7 +...
where
log(A(x)) = (1 + x + 2^2*x^2)/A(x) * x +
(1 + 2^2*x + 5^2*x^2 + 4^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 3^2*x + 9^2*x^2 + 13^2*x^3 + 18^2*x^4 + 12^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 4^2*x + 14^2*x^2 + 28^2*x^3 + 49^2*x^4 + 56^2*x^5 + 56^2*x^6 + 32^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 5^2*x + 20^2*x^2 + 50^2*x^3 + 105^2*x^4 + 161^2*x^5 + 210^2*x^6 + 200^2*x^7 + 160^2*x^8 + 80^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + x + 2*x^2)^n - see triangle A084600.

Crossrefs

Cf. A084600, A251688, A251689, A200537.

Programs

  • PARI
    /* By Definition: */
{a(n,p=1,q=1,r=2)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((p + q*x + r*x^2 +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, 50, print1(a(n), ", "))
  • PARI
    /* By G.F. Identity (faster): */
{a(n,p=1,q=1,r=2)=polcoeff( (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / ((1-p*r*x^2)^2 +x*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: (1 + x)*(1 + 4*x^3)*(1 - 3*x^2 + 4*x^4) / (1 - 2*x^2)^2.

A251689 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,2*n-k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,2*n-k) is the coefficient of x^k in (2+3*x+x^2)^n.

Original entry on oeis.org

1, 4, 9, 37, 40, 153, 144, 468, 432, 1260, 1152, 3168, 2880, 7632, 6912, 17856, 16128, 40896, 36864, 92160, 82944, 205056, 184320, 451584, 405504, 986112, 884736, 2138112, 1916928, 4608000, 4128768, 9879552, 8847360, 21086208, 18874368, 44826624, 40108032, 94961664, 84934656, 200540160
Offset: 0

Views

Author

Paul D. Hanna, Feb 23 2015

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 9*x^2 + 37*x^3 + 40*x^4 + 153*x^5 + 144*x^6 +...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (2^2 + 3^2*x + x^2)/A(x) * x +
(4^2 + 12^2*x + 13^2*x^2 + 6^2*x^3 + x^4)/A(x)^2 * x^2/2 +
(8^2 + 36^2*x + 66^2*x^2 + 63^2*x^3 + 33^2*x^4 + 9^2*x^5 + x^6)/A(x)^3 * x^3/3 +
(16^2 + 96^2*x + 248^2*x^2 + 360^2*x^3 + 321^2*x^4 + 180^2*x^5 + 62^2*x^6 + 12^2*x^7 + x^8)/A(x)^4 * x^4/4 +
(32^2 + 240^2*x + 800^2*x^2 + 1560^2*x^3 + 1970^2*x^4 + 1683^2*x^5 + 985^2*x^6 + 390^2*x^7 + 100^2*x^8 + 15^2*x^9 + x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients A200536(n,2*n-k) in (2+3*x+x^2)^n.

Crossrefs

Cf. A200537, A200536, A251688.

Programs

  • PARI
    {a(n)=polcoeff( (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / ((1-2*x^2)^2 +x*O(x^n)), n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((2+3*x+x^2+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),", "))

Formula

G.f.: (1+4*x)*(1+4*x^2)*(1+x^2)*(1+x^3) / (1-2*x^2)^2.

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

Views

Author

Paul D. Hanna, Mar 04 2015

Keywords

Comments

Compare the definition of this sequence to G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ) where T(n,k) = [x^k] (1+x+x^2)^n, which is satisfied by the rational function: G(x) = (1+x^3)*(1+x^6)/((1-x)*(1-x^4)).

Examples

			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.

Links

Crossrefs

Cf. A255839, A200537, A251687, A251688, A005721.

Programs

  • PARI
    /* 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

Views

Author

Paul D. Hanna, Mar 07 2015

Keywords

Comments

Compare the definition of this sequence to G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} binomial(2*n,k)^2 * x^k] / G(x)^n * x^n/n ), which is satisfied by the rational function: G(x) = (1+x^2)^2*(1+x^3)/((1-x)*(1-x^2)).

Examples

			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).

Links

Crossrefs

Cf. A248876, A200537, A251687, A251688, A005721.

Programs

  • PARI
    /* 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 1-4 of 4 results.

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