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 1-9 of 9 results.

A050383 Permutation rooted trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 25, 77, 262, 897, 3208, 11658, 43243, 162477, 618219, 2374699, 9200541, 35903017, 140997527, 556798525, 2209685939, 8807924914, 35248187347, 141564134395, 570402287162, 2305138038036, 9340981510156, 37946616550787
Offset: 1

Views

Author

Christian G. Bower, Nov 15 1999

Keywords

Links

Crossrefs

Cf. A004111, A005355, A091865, A308368.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = 1/Product[1 - x^n A[x^n], {n, 1, m}] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 02 2019 *)
  • 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)} /* Paul D. Hanna */

Formula

G.f. (with offset 0) satisfies: A(x) = 1/Product_{n>=1} (1 - x^n*A(x^n)). - Paul D. Hanna, Sep 28 2011
Shifts left under transform T where Ta is EULER(CIK(a)).
a(n) ~ c * d^n / n^(3/2), where d = 4.313133937842504228... and c = 0.153549235191409889... - Vaclav Kotesovec, Nov 05 2021

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).

A206301 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} A(x^k).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 453, 1003, 2200, 4860, 10681, 23552, 51819, 114186, 251326, 553634, 1218857, 2684461, 5910729, 13016952, 28662693, 63120135, 138991543, 306076520, 673995311, 1484205869, 3268315926, 7197126602, 15848588048, 34899932674
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 43*x^6 + 93*x^7 +...
such that, by definition,
A(x) = 1 + x*A(x) + x^2*A(x)*A(x^2) + x^3*A(x)*A(x^2)*A(x^3) + x^4*A(x)*A(x^2)*A(x^3)*A(x^4) + x^5*A(x)*A(x^2)*A(x^3)*A(x^4)*A(x^5) +...
The coefficients in Product_{k=1..n} A(x^k) begin:
n=2: [1, 1, 3, 5, 13, 25, 60, 124, 285, 609, 1369, 2970, 6611, ...];
n=3: [1, 1, 3, 6, 14, 28, 67, 139, 316, 683, 1523, 3317, 7369, ...];
n=4: [1, 1, 3, 6, 15, 29, 70, 145, 332, 713, 1596, 3468, 7717, ...];
n=5: [1, 1, 3, 6, 15, 30, 71, 148, 338, 728, 1627, 3540, 7868, ...];
n=6: [1, 1, 3, 6, 15, 30, 72, 149, 341, 734, 1642, 3570, 7941, ...];
n=7: [1, 1, 3, 6, 15, 30, 72, 150, 342, 737, 1648, 3585, 7971, ...]; ...

Crossrefs

Cf. A206302, A091865.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, subst(A, x, x^k +x*O(x^n))))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1+x*A(x) - x*A(x^2)/(1+x*A(x^2) - x*A(x^3)/(1+x*A(x^3) -...)))), a recursive continued fraction.

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

A308380 E.g.f. A(x) satisfies: A(x) = x * Product_{k>=1} (1 + A(x^k))^(1/k).

Original entry on oeis.org

1, 2, 9, 56, 455, 4224, 48391, 609104, 8814753, 140512400, 2483071481, 47387543928, 989622741367, 22107721563368, 530909919285495, 13581037512256544, 369627228319635329, 10633498287935101920, 323389433072136213289, 10342303284390333962600, 347514522157550224614711
Offset: 1

Views

Author

Ilya Gutkovskiy, May 23 2019

Keywords

Crossrefs

Cf. A091865, A308370, A308379.

Programs

  • Mathematica
    terms = 21; A[] = 0; Do[A[x] = x Product[(1 + A[x^k])^(1/k), {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]! // Rest

Formula

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

A329802 G.f. A(x) satisfies: A(x) = 1 / (1 - x * Product_{k>=1} A(x^k)).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 219, 777, 2803, 10315, 38496, 145516, 555764, 2142060, 8320207, 32538518, 128012533, 506300507, 2011932479, 8028941336, 32163411045, 129291553211, 521372223648, 2108522273338, 8549844313915, 34753397386201, 141584261960345
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2019

Keywords

Crossrefs

Cf. A050383, A091865.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = 1/(1 - x Product[A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Showing 1-9 of 9 results.

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

Content is available under The OEIS End-User License Agreement.