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.

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A385617 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(2*x)) ).

Original entry on oeis.org

1, 2, 10, 82, 1062, 22646, 846570, 58644858, 7808479582, 2038568219422, 1054007965984050, 1084591195956246130, 2226674324358059364150, 9131600163886719149539590, 74851744440590132840318820090, 1226745312860243142951267683147178, 40204124737879503807503331117931168974
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2025

Keywords

Crossrefs

Cf. A000051, A015083, A047749, A385618, A385622.

Programs

  • Mathematica
    terms = 17; A[] = 1; Do[A[x] = 1/( 1 - x*(A[x] + A[2*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+1)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k+1) * a(k) * a(n-1-k).
a(n) ~ c * 2^(n*(n-1)/2), where c = 30.250837358072598377515060923766952434821313428993180484... - Vaclav Kotesovec, Jul 05 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

A348875 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(2*x)).

Original entry on oeis.org

1, 2, 8, 56, 656, 13184, 477248, 32524928, 4295916032, 1117098857984, 576442191401984, 592587279827787776, 1215991461595100598272, 4985567391504232291377152, 40861715233637664786276712448, 669641809249948891254213657460736, 21945501536426419427607885034600595456
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A006318, A015083, A348876, A348877.

Programs

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

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).
a(n) ~ c * 2^(n*(n-1)/2), where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.51197587155650013108828169886454625305400323357646... - Vaclav Kotesovec, Nov 03 2021

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

A348878 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(2*x)).

Original entry on oeis.org

1, 1, 2, 5, 17, 74, 429, 3297, 34578, 495573, 9888497, 274123802, 10685538941, 583079000129, 44945515778914, 4867082587900837, 746167748281132753, 160981861948404281578, 49223569713040994430285, 21198824279482430844823713, 12946110661470835825027893426
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A001006, A015083, A348879, A348880.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[2^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} 2^k * a(k) * a(n-k-2).
a(n) ~ c * 2^(n*(n-2)/4), where c = 10.492153305884170498003413429333844276557493974205102819840538218355... - Vaclav Kotesovec, Nov 03 2021

A376095 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^2 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 5, 54, 983, 26863, 1029188, 52747686, 3491367091, 290276997159, 29639219057133, 3648073361410412, 532858993269296500, 91147584892512564076, 18051321652239427195456, 4098339933686479506696526, 1057506667415381878759070811, 307764793378228160791205354175
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015083, A051893, A088716, A376096, A376097.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 3 x^2 A[x] A'[x] + x^3 A[x] A''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

A348860 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(2*x))).

Original entry on oeis.org

1, 0, 1, 4, 37, 632, 20905, 1359692, 175426573, 45086173824, 23129393794129, 23707675064224020, 48577049664823958389, 199020196349510773741576, 1630572517436087330046884473, 26716930897552073378560239594588, 875487110213852689248519499248558685
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A005043, A015083, A348861, A348862.

Programs

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

Formula

a(n) = (-1)^n + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).
a(n) ~ c * 2^(n*(n-1)/2), where c = 0.658663398267275680037834076118178644268023291808559507713140088111498143... - Vaclav Kotesovec, Nov 02 2021
Previous Showing 31-40 of 45 results. Next

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