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 1-10 of 16 results. Next

A091865 G.f. satisfies A(x) = 1 + x*A(x)*A(x^2)*A(x^3)*...*A(x^n)*...

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 30, 57, 111, 214, 415, 798, 1547, 2983, 5765, 11132, 21510, 41528, 80231, 154940, 299280, 578017, 1116450, 2156280, 4164827, 8044023, 15536655, 30007988, 57958900, 111943844, 216213363, 417602892, 806575889, 1557852990
Offset: 0

Views

Author

Paul D. Hanna, Feb 05 2004

Keywords

Links

Crossrefs

Cf. A050383.

Programs

  • PARI
    {a(n)=local(A);A=1+x;for(i=1,n, A=1+x*prod(k=1,n,subst(A,x,x^k))+x*O(x^n)); polcoeff(A,n,x)}

Formula

G.f.: A(x) = 1 + x * Product_{k>=1} A(x^k).
a(n) ~ c * d^n, where d = 1.93144101206639993844275031844664... and c = 0.29686384114142059645291926346897... - Vaclav Kotesovec, Oct 02 2020

A196192 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x^n)^2).

Original entry on oeis.org

1, 1, 4, 16, 77, 389, 2128, 12019, 70185, 418788, 2544938, 15687842, 97871618, 616729500, 3919686231, 25096525793, 161723865118, 1048085548563, 6826585371618, 44664343473618, 293407529533947, 1934484748893113, 12796683165889635, 84906535878961845
Offset: 0

Views

Author

Paul D. Hanna, Sep 28 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 77*x^4 + 389*x^5 + 2128*x^6 +...
where
A(x) = 1/((1 - x*A(x)^2) * (1 - x^2*A(x^2)^2) * (1 - x^3*A(x^3)^2) *...).

Crossrefs

Cf. A050383, A196191.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*subst(A,x,x^k+x*O(x^n))^2))); polcoeff(A, n)}

A196191 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n/A(x^n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 4, 4, 6, 7, 7, 8, 10, 11, 12, 14, 17, 18, 20, 22, 26, 30, 31, 37, 39, 46, 49, 54, 61, 69, 74, 82, 91, 100, 114, 119, 136, 149, 159, 176, 193, 214, 227, 255, 276, 303, 324, 360, 394, 420, 462, 496, 548, 590, 638, 692, 749, 812, 874, 946, 1035, 1115, 1191, 1292, 1395, 1503
Offset: 0

Views

Author

Paul D. Hanna, Sep 28 2011

Keywords

Comments

The rate of growth of this sequence is surprisingly slow.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + 3*x^7 + 2*x^8 +...
where
A(x) = 1/((1 - x/A(x)) * (1 - x^2/A(x^2)) * (1 - x^3/A(x^3)) *...).

Links

Crossrefs

Cf. A050383, A196192.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k/subst(A, x, x^k+x*O(x^n))))); polcoeff(A, n)}

A308369 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} 1/(1 - A(x^k))^k.

Original entry on oeis.org

1, 1, 4, 12, 41, 133, 485, 1752, 6677, 25809, 102130, 409532, 1665128, 6837348, 28333334, 118288386, 497120101, 2101181482, 8926401690, 38093403136, 163224292328, 701951448268, 3028792691947, 13108224143298, 56887750453404, 247512117880754, 1079421026637431
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308370, A308371, A308372.

Programs

  • Mathematica
    terms = 27; A[] = 0; Do[A[x] = x Product[1/(1 - A[x^k])^k, {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} Sum_{d|k} d^2 * A(x^d)^(k/d) / k).

A308370 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + A(x^k))^k.

Original entry on oeis.org

1, 1, 3, 8, 20, 47, 118, 280, 681, 1640, 3963, 9523, 23004, 55377, 133477, 321597, 775054, 1867304, 4499934, 10842847, 26127768, 62958232, 151708512, 365562567, 880881465, 2122617010, 5114772619, 12324827128, 29698572295, 71563264162, 172442689864, 415527172616
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308369, A308371, A308372.

Programs

  • Mathematica
    terms = 32; A[] = 0; Do[A[x] = x Product[(1 + A[x^k])^k, {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(-Sum_{k>=1} Sum_{d|k} d^2 * (-A(x^d))^(k/d) / k).

A308371 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} 1/(1 - k*A(x^k)).

Original entry on oeis.org

1, 1, 4, 12, 42, 135, 500, 1797, 6885, 26612, 105561, 423734, 1726531, 7101261, 29486169, 123341520, 519422274, 2199966624, 9365714175, 40052639066, 171985425594, 741214499791, 3205096564624, 13901238793616, 60460193311425, 263627546862787, 1152207975128287
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308369, A308370, A308372.

Programs

  • Mathematica
    terms = 27; A[] = 0; Do[A[x] = x Product[1/(1 - k A[x^k]), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} Sum_{d|k} d * (d * A(x^d))^(k/d) / k).

A308372 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + k*A(x^k)).

Original entry on oeis.org

1, 1, 3, 8, 19, 45, 110, 259, 614, 1466, 3479, 8239, 19581, 46445, 110209, 261555, 620649, 1472597, 3494663, 8292514, 19677729, 46694303, 110804310, 262932172, 623928374, 1480555791, 3513297447, 8336903884, 19783134767, 46944538382, 111397439864, 264341463510
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Crossrefs

Cf. A050383, A091865, A308369, A308370, A308371.

Programs

  • Mathematica
    terms = 32; A[] = 0; Do[A[x] = x Product[(1 + k A[x^k]), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * exp(-Sum_{k>=1} Sum_{d|k} d * (-d * A(x^d))^(k/d) / k).

A005355 Number of asymmetric permutation rooted trees with n nodes.

Original entry on oeis.org

0, 1, 1, 1, 3, 7, 21, 61, 187, 577, 1825, 5831, 18883, 61699, 203429, 675545, 2258291, 7592249, 25656477, 87096661, 296891287, 1015797379, 3487272317, 12008898531, 41471260883, 143588078449, 498343911529, 1733410858955, 6041795275027, 21098924740155
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A004111, A050383.

Programs

  • Mathematica
    a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[a[k]*a[n-k], {k, 1, n-1}] - If[EvenQ[n-1], a[(n-1)/2], 0]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 04 2016 *)

Formula

Shifts left under transform T where Ta has g.f. (1-A(x^2))/(1-A(x)).

Extensions

More terms, formula from Christian G. Bower, Nov 15 1999

A205772 G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - 2*x^n*A(x^n)).

Original entry on oeis.org

1, 2, 10, 50, 290, 1766, 11442, 76522, 526574, 3697722, 26403186, 191072922, 1398344838, 10330855286, 76945148882, 577135722754, 4355579825058, 33050011129198, 251996066644866, 1929712025078322, 14834772898730766, 114445491235869774
Offset: 0

Views

Author

Paul D. Hanna, Jan 31 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 290*x^4 + 1766*x^5 +...,
where A(x) = 1/((1 - 2*x*A(x)) * (1 - 2*x^2*A(x^2)) * (1 - 2*x^3*A(x^3)) * ...).

Crossrefs

Cf. A050383, A196192, A205773.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-2*x^k*subst(A, x, x^k+x*O(x^n))))); polcoeff(A, n)}

A308368 G.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + A(x^k))/(1 - A(x^k)).

Original entry on oeis.org

1, 2, 8, 32, 142, 652, 3176, 15916, 82120, 432334, 2315360, 12569180, 69018212, 382630996, 2138788360, 12040391240, 68204335458, 388473940840, 2223439634504, 12781420672112, 73762215951860, 427196466303812, 2482105805258232, 14464061008937328, 84514482402557528
Offset: 1

Views

Author

Ilya Gutkovskiy, May 22 2019

Keywords

Links

Crossrefs

Cf. A050383, A091865.

Programs

  • Mathematica
    terms = 25; A[] = 0; Do[A[x] = x Product[(1 + A[x^k])/(1 - A[x^k]), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

Formula

a(n) ~ c * d^n / n^(3/2), where d = 6.218062815147882349... and c = 0.1489003353315039... - Vaclav Kotesovec, Nov 05 2021
Showing 1-10 of 16 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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