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.

Previous Showing 11-20 of 27 results. Next

A366356 G.f. satisfies A(x) = 1/(1 - x) + x/A(x).

Original entry on oeis.org

1, 2, -1, 6, -17, 71, -292, 1284, -5807, 26961, -127627, 613815, -2990680, 14730714, -73229290, 366936232, -1851352819, 9397497759, -47957377933, 245903408245, -1266266092111, 6545667052321, -33954266444497, 176689391245147, -922112642288148, 4825154135801698
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Links

Crossrefs

Partial sums give A363816.
Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366357, A366358, A366359.
Cf. A112478, A366363.

Programs

  • Mathematica
    A366356[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[2k-1,n-k]/(2k-1),{k,0,n}];
Array[A366356,30,0] (* Paolo Xausa, Oct 20 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(2*k-1, n-k)/(2*k-1));

Formula

G.f.: A(x) = -2*x*(1-x) / (1-sqrt(1+4*x*(1-x)^2)).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(2*k-1,n-k)/(2*k-1).

A349533 G.f. A(x) satisfies A(x) = 1 / ((1 - 2 * x) * (1 - x * A(x)^2)).

Original entry on oeis.org

1, 3, 13, 74, 499, 3719, 29494, 243888, 2078431, 18122369, 160885449, 1449268478, 13213370392, 121696581804, 1130565483472, 10581614352704, 99685591788687, 944490400760597, 8994266558594671, 86040075341770806, 826423263373253923, 7967095415955791687
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2021

Keywords

Links

Crossrefs

Cf. A001764, A064613, A199475, A349253, A349534, A349535.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = 1/((1 - 2 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 2^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 2^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 21}]

Formula

a(n) = 2^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 2^(n-k) / (2*k+1).
a(n) = 2^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^3/2^5), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021
a(n) ~ 177^(1/4) * (43 + 3*sqrt(177))^(n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 22 2021

A349534 G.f. A(x) satisfies A(x) = 1 / ((1 - 3 * x) * (1 - x * A(x)^2)).

Original entry on oeis.org

1, 4, 21, 138, 1063, 9075, 82770, 789204, 7766721, 78267306, 803447526, 8371413999, 88300495746, 941004684748, 10116276976218, 109578418285452, 1194764348642313, 13102287157827918, 144422108994233625, 1599198859915070640, 17780781456147340764
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2021

Keywords

Links

Crossrefs

Cf. A001764, A104455, A199475, A349254, A349533, A349535.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/((1 - 3 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 3^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 3^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 20}]

Formula

a(n) = 3^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 3^(n-k) / (2*k+1).
a(n) = 3^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^2/2^4), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021
a(n) ~ sqrt(5/Pi) * 3^(n-1) * 4^n / n^(3/2). - Vaclav Kotesovec, Nov 22 2021

A349535 G.f. A(x) satisfies A(x) = 1 / ((1 - 4 * x) * (1 - x * A(x)^2)).

Original entry on oeis.org

1, 5, 31, 232, 2007, 19213, 196912, 2114360, 23467039, 266945359, 3094963919, 36433302124, 434269638196, 5230582604476, 63560612650120, 778281652963504, 9593248679529551, 118938874458415987, 1482248642350235245, 18557299563105608336
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 21 2021

Keywords

Links

Crossrefs

Cf. A001764, A104498, A199475, A349533, A349534.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/((1 - 4 x) (1 - x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 4^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n + k, 2 k] Binomial[3 k, k] 4^(n - k)/(2 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = 4^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,2*k) * binomial(3*k,k) * 4^(n-k) / (2*k+1).
a(n) = 4^n*F([1/3, 2/3, -n, 1+n], [1/2, 1, 3/2], -3^3/2^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 21 2021
a(n) ~ 273^(1/4) * (59 + 3*sqrt(273))^(n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 22 2021

A366357 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^2.

Original entry on oeis.org

1, 2, -3, 19, -105, 690, -4781, 34708, -260189, 1999169, -15660175, 124596499, -1004110947, 8179379808, -67239070867, 557098881920, -4647368670949, 39001655222788, -329048378867467, 2789241880512899, -23743798316713367, 202894843070927860
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366358, A366359.
Cf. A364393, A366326, A366364.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(3*k-1, k)*binomial(3*k-1, n-k)/(3*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k-1,k) * binomial(3*k-1,n-k)/(3*k-1).

A366358 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^3.

Original entry on oeis.org

1, 2, -5, 40, -319, 2908, -28151, 284908, -2977115, 31875709, -347884084, 3855802690, -43283239649, 491083601339, -5622489637406, 64877058557080, -753705528179423, 8808460811302729, -103487549564845199, 1221565052783161764, -14480208437556590345
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366357, A366359.
Cf. A364407, A366327, A366365.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(4*k-1, k)*binomial(4*k-1, n-k)/(4*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(4*k-1,k) * binomial(4*k-1,n-k)/(4*k-1).

A366359 G.f. satisfies A(x) = 1/(1 - x) + x/A(x)^4.

Original entry on oeis.org

1, 2, -7, 69, -715, 8351, -103735, 1346247, -18035023, 247520970, -3462344959, 49181268701, -707502644111, 10286493363184, -150913708053635, 2231345941617611, -33215679733509159, 497392118745778015, -7487512016559918595, 113242852989349372915
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A007317, A199475, A349289, A349290, A349291, A349292, A349293, A366356, A366357, A366358.
Cf. A364408, A366328, A366366.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(5*k-1, n-k)/(5*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(5*k-1,n-k)/(5*k-1).

A379284 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x)^4)).

Original entry on oeis.org

1, 2, 15, 158, 1943, 26099, 371128, 5491868, 83692617, 1304579981, 20703125143, 333366138381, 5433036837372, 89448269251685, 1485469625972490, 24854484773368344, 418581393456669989, 7090045259711970090, 120706208890692261466, 2064356606197948427512, 35449776962011108029539
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A118969, A199475, A379209, A379285, A379286, A379287.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(3*n+k, n-k)/(2*n+3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(3*n+k,n-k)/(2*n+3*k+1).

A379287 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x)^6)).

Original entry on oeis.org

1, 2, 19, 268, 4477, 82110, 1597963, 32402460, 677152153, 14481799261, 315417278757, 6972246638416, 156017257712825, 3527275634678216, 80447862652931941, 1848737311902300600, 42766087499793329349, 995043161703028219128, 23271045049097437148389
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2024

Keywords

Crossrefs

Cf. A118969, A199475, A379209, A379284, A379285, A379286.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(2*n+5*k+1, k)*binomial(3*n+3*k, n-k)/(2*n+5*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(2*n+5*k+1,k) * binomial(3*n+3*k,n-k)/(2*n+5*k+1).

A366176 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3/(1 - x)^2.

Original entry on oeis.org

1, 1, 5, 27, 161, 1030, 6921, 48190, 344669, 2517303, 18695908, 140771477, 1072130229, 8244820518, 63931532190, 499308229278, 3924204043333, 31012883225891, 246304580923299, 1964794017165157, 15735626383151876, 126476316316459089, 1019883740031357941
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Partial sums give A199475.
Cf. A001764, A213282, A307678.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+k-1, n-k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+k-1,n-k) * binomial(3*k,k)/(2*k+1).
Previous Showing 11-20 of 27 results. Next

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

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