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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A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 14, 1, 1, 1, 5, 43, 171, 42, 1, 1, 1, 6, 89, 1252, 3113, 132, 1, 1, 1, 7, 161, 5885, 104098, 106419, 429, 1, 1, 1, 8, 265, 20466, 1518897, 25511272, 7035649, 1430, 1, 1, 1, 9, 407, 57799, 12833546, 1558435125, 18649337311, 915028347, 4862, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2017

Keywords

Comments

This is the transpose of the array in A090182.

Examples

			G.f. of column k: A_k(x) = 1 + x + (k + 1)*x^2 + (k^3 + k^2 + 2*k + 1)*x^3 + (k^6 + k^5 + 2*k^4 + 3*k^3 + 3*k^2 + 3*k + 1)*x^4 + ...
Square array begins:
  1,   1,     1,       1,        1,         1,  ...
  1,   1,     1,       1,        1,         1,  ...
  1,   2,     3,       4,        5,         6,  ...
  1,   5,    17,      43,       89,       161,  ...
  1,  14,   171,    1252,     5885,     20466,  ...
  1,  42,  3113,  104098,  1518897,  12833546,  ...

Links

Crossrefs

Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096.
Main diagonal gives A290777.
Cf. A090182, A290789.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    Table[Function[k, SeriesCoefficient[1/(1 - x/(1 + ContinuedFractionK[-k^i x, 1, {i, 1, n}])), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
  • Python
    from sympy.core.cache import cacheit
@cacheit
def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*k**j for j in range(n))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 10 2017, after Maple code

Formula

G.f. of column k: 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))), a continued fraction.

A090182 Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 17, 4, 1, 1, 1, 42, 171, 43, 5, 1, 1, 1, 132, 3113, 1252, 89, 6, 1, 1, 1, 429, 106419, 104098, 5885, 161, 7, 1, 1, 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1, 1, 4862, 915028347, 18649337311, 1558435125, 12833546, 57799, 407, 9, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 20 2004, Oct 16 2008

Keywords

Examples

			Triangle begins:
  1;
  1,    1;
  1,    1,       1;
  1,    2,       1,        1;
  1,    5,       3,        1,       1;
  1,   14,      17,        4,       1,     1;
  1,   42,     171,       43,       5,     1,   1;
  1,  132,    3113,     1252,      89,     6,   1, 1;
  1,  429,  106419,   104098,    5885,   161,   7, 1, 1;
  1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1;
This sequence formatted as a square array:
  1, 1, 1,   1,     1,        1,           1,               1, ...
  1, 1, 2,   5,    14,       42,         132,             429, ...
  1, 1, 3,  17,   171,     3113,      106419,         7035649, ...
  1, 1, 4,  43,  1252,   104098,    25511272,     18649337311, ...
  1, 1, 5,  89,  5885,  1518897,  1558435125,   6386478643785, ...
  1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, ...

Links

Crossrefs

The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.
T(2n,n) gives A290777.
Cf. A290759.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k=n, 1, add(
      T(j+k, k)*T(n-j-1, k)*k^j, j=0..n-k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    nmax = 10; col[k_] := col[k] = Module[{A}, A[] = 0; Do[A[x] = Normal[1/(1 - x*A[k*x]) + O[x]^(nmax-k+1)], {nmax-k+1}]; CoefficientList[A[x], x]];
T[n_, k_] := col[k][[n-k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, using g.f. given for column sequences *)

A385526 E.g.f. A(x) satisfies A(x) = exp(x*A(3*x)).

Original entry on oeis.org

1, 1, 7, 208, 23365, 9588976, 14040296659, 71747056999360, 1255862559932597257, 74168744207577385109248, 14599375893944236344767578111, 9483024632344097320792984610415616, 20158786175666520486280070249843236771213, 139271933359690469686747131442731382830399594496
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2025

Keywords

Crossrefs

Cf. A000272, A096538, A141369, A385527, A385528, A385529, A385530.
Cf. A015084.

Programs

  • Mathematica
    nmax = 15; A[] = 1; Do[A[x] = E^(x*A[3*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)
  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385526(n)
  A(3, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * 3^k * binomial(n-1,k) * a(k) * a(n-1-k).
a(n) ~ c * n! * 3^(n*(n-1)/2), where c = 1.361839192264541770366149558100099215697354561... - Vaclav Kotesovec, Jul 02 2025

A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

Original entry on oeis.org

1, 2, 9, 103, 3101, 261192, 64285189, 47059492688, 103060910397021, 676492249628112382, 13317427360663454672669, 786420726604930579016189223, 139314431838014895142151741877241, 74037818920801629179455290512454633872, 118040419689979917511971388549088825283510249
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A007317, A015084, A348857, A348859.

Programs

  • Mathematica
    nmax = 14; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[3 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

Formula

a(n) = 1 + Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).
a(n) ~ c * 3^(n*(n-1)/2), where c = 4.508135635010167805309616576501854361005320931661829410476785686203732753... - Vaclav Kotesovec, Nov 02 2021

A348188 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(3*x)).

Original entry on oeis.org

1, 1, 7, 139, 7813, 1282741, 626077507, 914089078999, 4000061058178633, 52496811551448519241, 2066694521388276020211487, 244076623554395367965602542499, 86475371441574361841467969073397133, 91913288701991663661449175594278601481981
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 03 2021

Keywords

Crossrefs

Cf. A001003, A015084, A348901, A348902.

Programs

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

Formula

a(0) = 1; a(n) = -a(n-1) + 2 * Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).
a(n) ~ c * 3^(n*(n-1)/2) * 2^n, where c = 0.68317332785969015770364424102230433743028917778042859282957908502822... - Vaclav Kotesovec, Nov 03 2021

A348876 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(3*x)).

Original entry on oeis.org

1, 2, 10, 122, 3778, 321794, 79518154, 58289895290, 127713856067074, 838441945709583746, 16506407616569722560778, 974752895709158578160969978, 172678450359956040815290930278850, 91769099059347441553324620759011469698, 146309952397373808216450794120154608358754762
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A006318, A015084, A348875, A348877.

Programs

  • Mathematica
    nmax = 14; A[] = 0; Do[A[x] = 1/(1 - x - x A[3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).
a(n) ~ c * 3^(n*(n-1)/2), where c = 2*Product_{j>=1} (3^j+1)/(3^j-1) = QPochhammer(-1, 1/3) / QPochhammer(1/3) = 5.58779203552209791475992929265... - Vaclav Kotesovec, Nov 03 2021

A348879 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(3*x)).

Original entry on oeis.org

1, 1, 2, 6, 29, 221, 2815, 59607, 2175115, 134785987, 14543011028, 2682224473296, 864129873439979, 476879023670530355, 460188677448639450646, 761220053428592181980874, 2202591080616789155249254723, 10927081698418028875550581480027, 94836180093445711611212497662570806
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A001006, A015084, A348878, A348880.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[3^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} 3^k * a(k) * a(n-k-2).
a(n) ~ c * 3^(n*(n-2)/4), where c = 4.2101130581370834571021724998929772199905440992108887037121562184404379... - Vaclav Kotesovec, Nov 03 2021

A376096 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 9, 260, 17215, 2189997, 477731884, 164858203944, 84745577983095, 61951785517193675, 62077057930391945969, 82749694746019635920952, 143157935882304543684640676, 314805573970543375502985796300, 864458294787075036217714712292600, 2919280453922974335841433174057739408
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015084, A088716, A256019, A376095, A376097.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 7 x^2 A[x] A'[x] + 6 x^3 A[x] A''[x] + x^4 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 + 7 * x^2 * A(x) * A'(x) + 6 * x^3 * A(x) * A''(x) + x^4 * A(x) * A'''(x).

A348861 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(3*x))).

Original entry on oeis.org

1, 0, 1, 9, 253, 20754, 5064301, 3696964686, 8088964208893, 53079786931826952, 1044822534120774924517, 61696770693051062357722413, 10929459535778338593167921597497, 5808373834674826377471903826388912406, 9260429805605038398327449254849233420999649
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A005043, A015084, A348860, A348862.

Programs

  • Mathematica
    nmax = 14; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[3 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

Formula

a(n) = (-1)^n + Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).
a(n) ~ c * 3^(n*(n-1)/2), where c = 0.353669086629957226916356822657293915192042094460583129054709983118948... - Vaclav Kotesovec, Nov 02 2021

A352007 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 3^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 4, 7, 19, 40, 178, 379, 1237, 2941, 10378, 24628, 78928, 198820, 813550, 1971907, 6587245, 16980079, 61488286, 155573011, 515316037, 1363261084, 4937498686, 12796438252, 42078038668, 113153315824, 390012381346, 1036020692356, 3379994401042, 9240830253940
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2022

Keywords

Crossrefs

Cf. A000621, A015084, A352006, A352008.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[3^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x A[3 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(3*x^2)).
Previous Showing 21-30 of 33 results. Next

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