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 10 results.

A171200 G.f. satisfies A(x) = 1 + x*A(2x)^3.

Original entry on oeis.org

1, 1, 6, 84, 2312, 121056, 12173568, 2391143424, 928316362752, 716762538541056, 1103851068987015168, 3395472896229407981568, 20875407961847891162038272, 256600638160251032545689337856, 6307244441266548036155317187248128
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171201, A171202-A171211, A343439.

Programs

  • Mathematica
    m = 15; A[] = 0; Do[A[x] = 1 + x A[2x]^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^3); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Nov 03 2021

A135868 G.f. A(x) = (1 + x*A(2*x))^2.

Original entry on oeis.org

1, 2, 9, 80, 1368, 45344, 2952208, 381005824, 97928967168, 50238812700672, 51495089926791168, 105513593089512439808, 432289346894817079853056, 3541746937174189008054386688, 58031524865557213409213540204544, 1901635048957282075182110362797342720, 124627456377710926240825658971690792648704
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Comments

Self-convolution of A135867 such that A135867(n+1) = 2^n*a(n).

Crossrefs

Cf. A135867.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = (1 + x*A[2x])^2 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)^2);polcoeff(A^2,n)}

Extensions

a(15)-a(16) from Stefano Spezia, Apr 02 2025

A171202 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^4.

Original entry on oeis.org

1, 1, 8, 152, 5664, 399376, 53846016, 14141384704, 7330134466560, 7551251740344320, 15510852680588984320, 63626087316632048238592, 521607805205244557347782656, 8549156556447111748331767857152, 280190094729160875643888549840814080, 18364219805837823940403573170370661842944
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200, A171201, A171203, A171204-A171211, A343439.
Cf. A143047, A171193.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1 + x*A[2x]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^4); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

A171204 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^5.

Original entry on oeis.org

1, 1, 10, 240, 11280, 1000080, 169100832, 55605632640, 36058105605120, 46450803286978560, 119290436529298554880, 611727201854914747760640, 6268994998754867059071385600, 128439243721180540266999017635840, 5261899692949082390205726962630000640, 431096933496167311430326245852780460769280
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200-A171203, A171205, A171206-A171211, A343439.
Cf. A143048, A171194.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1 + x*A[2x]^5 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^5); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

A171206 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^6.

Original entry on oeis.org

1, 1, 12, 348, 19744, 2108784, 428817600, 169398274624, 131889504749568, 203937600707475456, 628561895904796999680, 3868208404121906515820544, 47571342639450113377565933568, 1169589733863427138021074362433536, 57499379103783344787572704263568097280, 5652994168279651703590653986228287051923456
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200-A171205, A171207, A171208-A171211, A343439.
Cf. A143049, A171195.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1+x*A[2x]^6 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^6); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 2025

A171208 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^7.

Original entry on oeis.org

1, 1, 14, 476, 31640, 3953488, 939383200, 433281169216, 393718899904640, 710399428248892928, 2554705943898166145024, 18342976469146094416494592, 263185684727811758287894478848, 7549222852919288301041224694890496, 432993292623369448352459156263293419520
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200-A171207, A171209, A171210-A171211, A343439.
Cf. A171196.

Programs

  • Mathematica
    terms = 15; A[] = 0; Do[A[x] = 1+x*A[2x]^7 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^7); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

A171210 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^8.

Original entry on oeis.org

1, 1, 16, 624, 47552, 6804576, 1849952000, 975746615040, 1013611906401280, 2090459909088346368, 8592166589474459877376, 70508055994868618069409792, 1156194054760373598022278840320, 37902377449956182566891283844956160, 2484501232375923934830943089632156319744
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A135867, A171200-A171209, A171211, A343439.
Cf. A171197.

Programs

  • Mathematica
    terms = 15; A[] = 0; Do[A[x] = 1+x*A[2x]^8 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A, x, 2*x)^8); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 08 2025

A135869 G.f. A(x) = 1 + x*A(3x)^2.

Original entry on oeis.org

1, 1, 6, 117, 6642, 1097874, 537135948, 784812995973, 3435153688724346, 45086429284345043334, 1775007791598340247784372, 209630197234751724563143145346, 74271350069687203431923556331222068
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Comments

Self-convolution equals A135870.

Crossrefs

Cf. A135867, A135870.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,3*x)^2);polcoeff(A,n)}

A349012 G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x * A(x)).

Original entry on oeis.org

1, 2, 8, 52, 552, 10208, 350112, 23159760, 3012389984, 777296223040, 399542726439936, 409933997609848192, 840366306053838941952, 3443822768422065940362240, 28218687132517064788995222528, 462391421142204650963524251763968, 15152566983759983965941543133445666304
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

Crossrefs

Cf. A006318, A047891, A135867, A348875, A349013.

Programs

  • Mathematica
    nmax = 16; A[] = 0; Do[A[x] = (1 + x A[2 x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2^(n - 1) a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = 2^(n-1) * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
a(n) ~ c * 2^(n*(n-1)/2), where c = 11.40022022373995418911523299051117421707893086825818379118899572625286143... - Vaclav Kotesovec, Nov 06 2021

A349013 G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x^2 * A(x)).

Original entry on oeis.org

1, 1, 3, 14, 119, 1938, 62291, 3990822, 510954584, 130812494690, 66977027614263, 68584738950054184, 140461679588587238360, 575331176899085656611498, 4713113282218244019907299347, 77219649166807400596911172015640, 2530333473325322974256848862067093128
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 05 2021

Keywords

Crossrefs

Cf. A000108, A001006, A135867, A348878, A349012.

Programs

  • Mathematica
    nmax = 16; A[] = 0; Do[A[x] = (1 + x A[2 x])/(1 - x^2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = 2^(n - 1) a[n - 1] + Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = 2^(n-1) * a(n-1) + Sum_{k=0..n-2} a(k) * a(n-k-2).
a(n) ~ c * 2^(n*(n-1)/2), where c = 1.903611332661849505687584024897203124045921097899485586512432930506328189... - Vaclav Kotesovec, Nov 06 2021
Showing 1-10 of 10 results.

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

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