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 1-4 of 4 results.

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4*k*x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 07 2017

Keywords

Comments

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

Examples

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

Programs

  • Maple
    ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019
  • Mathematica
    Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

A292784 a(n) = n! * [x^n] 1/sqrt(1 - 2*n*x).

Original entry on oeis.org

1, 1, 12, 405, 26880, 2953125, 484989120, 111289483305, 34007836262400, 13350287284158825, 6547290750000000000, 3922838769902739011325, 2819575386162274605465600, 2394486245934541921935898125, 2371947271643716575046318080000, 2710687260280640086154937744140625, 3539907755812512418187309922385920000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 23 2017

Keywords

Crossrefs

Main diagonal of A292783.
Cf. A000312, A001147, A061711, A291699.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/Sqrt[1 - 2 n x], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-i n x, 1, {i, 1, n}]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[n^n (2 n - 1)!!, {n, 1, 16}]]

Formula

a(n) = [x^n] 1/(1 - n*x/(1 - 2*n*x/(1 - 3*n*x/(1 - 4*n*x/(1 - 5*n*x/(1 - ...)))))), a continued fraction.
a(n) = A000312(n)*A001147(n).

A349639 a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * k^k.

Original entry on oeis.org

1, 2, 11, 163, 4177, 150606, 7002679, 399296682, 26997867705, 2112814307980, 187919721166951, 18727570061711897, 2067435790679136937, 250474099952311886236, 33043529154916822685459, 4715582224589290429430011, 723854564711343436767660481, 118933484485939500023357177356
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 23 2021

Keywords

Crossrefs

Cf. A007317, A064613, A291699, A292632, A349603, A349640.

Programs

  • Mathematica
    Table[1+Sum[Binomial[n, j]*CatalanNumber[j]*j^j, {j, 1, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * (binomial(2*k,k)/(k+1)) * k^k); \\ Michel Marcus, Nov 23 2021

Formula

a(n) ~ c * 2^(2*n) * n^(n - 3/2) /sqrt(Pi), where c = Sum_{k>=0} 1/(4^k*k!*exp(k)) = exp(exp(-1)/4) = 1.09633177846412646399584148732...

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.

Original entry on oeis.org

1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

Crossrefs

Cf. A006318, A064063, A135861, A291699, A292798, A365816, A366012, A366016, A366037.

Programs

  • Maple
    A366038 := proc(n)
    add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;
    %/(n+1) ;
end proc:
seq(A366038(n),n=0..80) ; # R. J. Mathar, Oct 24 2024
  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]

Formula

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).
a(n) ~ phi^(3*n + 3/2) * exp(n/phi^2 + 1/(2*phi)) * n^(n - 3/2) / (5^(1/4) * sqrt(2*Pi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Sep 27 2023
Showing 1-4 of 4 results.

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