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.

A135867 G.f. satisfies A(x) = 1 + x*A(2*x)^2.

Original entry on oeis.org

1, 1, 4, 36, 640, 21888, 1451008, 188941312, 48768745472, 25069815595008, 25722272102744064, 52730972085034156032, 216091838647321476726784, 1770657164881170759078117376, 29013990909330956353981535748096
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Comments

Self-convolution equals A135868 such that 2^n*A135868(n) = a(n+1) for n >= 0.

Links

Crossrefs

Cf. A015083, A135868, A171192.
Cf. A171200, A171202, A171204, A171206, A171208, A171210, A343439.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1 + x*A[2*x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 04 2021 *)
  • 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,n)}
  • PARI
    a(n)=if(n==0,1,2^(n-1)*sum(k=0,n-1,a(k)*a(n-k-1))) \\ Paul D. Hanna, Feb 09 2010

Formula

a(n) = 2^(n-1)*Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Feb 09 2010
a(n) ~ c * 2^(n*(n+1)/2), where c = 0.715337433614869740944075474484711589980951273610257702786245519231799678... - Vaclav Kotesovec, Nov 04 2021

A171198 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^8).

Original entry on oeis.org

1, 1, 17, 689, 53777, 7805201, 2138582801, 1132509669905, 1178804946216209, 2433551908785577745, 10007244528797884954897, 82140401194398306308608785, 1347106337625031145913841134865, 44163564651481078406730693648713489
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171197.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^8) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^8) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n+5)/2), where c = 0.265929653305627916979803234586945454418485... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_9>=0 and x_1+x_2+...+x_9=n-1} (1/2)^x_1 * Product_{k=1..9} a(x_k). - Seiichi Manyama, Jul 06 2025

A171193 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^3).

Original entry on oeis.org

1, 1, 7, 109, 3207, 174581, 17929279, 3559607005, 1389312382199, 1075527698708485, 1658535837898129263, 5105026337441341642861, 31395991691829167745766311, 385982564381552315528268500501
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192, A171194-A171198.
Cf. A168601.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^3) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^3) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 3^n, where c = 0.80142677004566734464115933731029720165641... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (1/2)^i * a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 06 2025

A171194 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^4).

Original entry on oeis.org

1, 1, 9, 185, 7241, 525513, 71973193, 19054326985, 9916177373001, 10235479554015689, 21045100094428458057, 86370025530284981044937, 708236082282948046820221257, 11609413456993946896013575994313
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192, A171193, A171195-A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^4) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^4) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n+3)/2), where c = 0.5726679317239416602436569686037310143000778... - Vaclav Kotesovec, Nov 03 2021
a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (1/2)^i * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 06 2025

A171195 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^5).

Original entry on oeis.org

1, 1, 11, 281, 13731, 1245601, 213268203, 70580511385, 45914883339027, 59241954299963729, 152258885235304955131, 781096727709105092232777, 8006263111571482684378716067, 164048440920655457493139473502081
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171194, A171196-A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^5) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^5) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 5^n, where c = 0.444871440417987089861554304425221691031547... - Vaclav Kotesovec, Nov 03 2021
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} (1/2)^x_1 * Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 06 2025

A171196 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^6).

Original entry on oeis.org

1, 1, 13, 397, 23261, 2532093, 520285021, 206632208765, 161306955003037, 249753449538341821, 770275887324912000733, 4741871606773351738426877, 58325180751309642789169099037, 1434100517517383561901937569640509
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171195, A171197, A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^6) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^6) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n+1)/2) * 3^n, where c = 0.363484431362432363073577975298028185297326... - Vaclav Kotesovec, Nov 03 2021
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} (1/2)^x_1 * Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 06 2025

A171197 G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^7).

Original entry on oeis.org

1, 1, 15, 533, 36415, 4624621, 1108685495, 513716588981, 467874135168079, 845152554936920445, 3041003426951554000167, 21840734269889733272106629, 313415404907854466274076819391, 8990640466019774671530066108827853
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Links

Crossrefs

Cf. A015083, A171192-A171196, A171198.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^7) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^7) ); polcoeff(A, n)}

Formula

a(n) ~ c * 2^(n*(n-1)/2) * 7^n, where c = 0.307176924551399606223470587229647816147018... - Vaclav Kotesovec, Nov 03 2021
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} (1/2)^x_1 * Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 06 2025

A385648 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x))^2 ).

Original entry on oeis.org

1, 4, 64, 1872, 91328, 7563648, 1115422976, 306988895488, 162926170881024, 169827391985854464, 350891899856754294784, 1443597302250006622052352, 11851990053153536620868173824, 194396568906445310993071164686336, 6373487768490075927307409156798611456
Offset: 0

Views

Author

Seiichi Manyama, Jul 06 2025

Keywords

Crossrefs

Cf. A385617, A385649.
Cf. A171192, A385650.

Formula

a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+j+k=n-1} (2^j+1) * (2^k+1) * a(i) * a(j) * a(k).

A349286 G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(3*x)^2).

Original entry on oeis.org

1, 1, 7, 148, 8695, 1455433, 715092412, 1046279830240, 4581706851060703, 60144130721142806611, 2367934845363738749478583, 279660212608573899258160179472, 99083333806943049370375436233999876, 105314387436569180750449877597844106742044
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Crossrefs

Cf. A001764, A015084, A171192, A349287.

Programs

  • Mathematica
    nmax = 13; A[] = 0; Do[A[x] = 1/(1 - x A[3 x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[3^(i + j) a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 13}]

Formula

a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 3^(i+j) * a(i) * a(j) * a(n-i-j-1).
a(n) ~ c * 3^(n*(n-1)/2) * 2^n, where c = 0.7827821536020754599551185976017878354967268590673601224489106938395... - Vaclav Kotesovec, Nov 13 2021

A349287 G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(4*x)^2).

Original entry on oeis.org

1, 1, 9, 321, 42937, 22259313, 45726174057, 374866565186721, 12285883413435994137, 1610409077693221284887505, 844327818646575560326075164105, 1770688839714867344554954935264852993, 14853625190589908388648838739441430566681721
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Crossrefs

Cf. A001764, A015085, A171192, A349286.

Programs

  • Mathematica
    nmax = 12; A[] = 0; Do[A[x] = 1/(1 - x A[4 x]^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[4^(i + j) a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 12}]

Formula

a(0) = 1; a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} 4^(i+j) * a(i) * a(j) * a(n-i-j-1).
a(n) ~ c * 2^(n^2), where c = 0.6660597482166910709619924328518595274303795046... - Vaclav Kotesovec, Nov 14 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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