A015085 -id:A015085 - cp's OEIS frontend

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 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).

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

Ilya Gutkovskiy, Aug 09 2017

This is the transpose of the array in A090182.

			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,  ...

Seiichi Manyama, Antidiagonals n = 0..55, flattened

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

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

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

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

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

Philippe Deléham, Jan 20 2004, Oct 16 2008

			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, ...

Alois P. Heinz, Rows n = 0..55, flattened
Lun Lv, Zhihong Liu, Some Identities Related to Restricted Lattice Paths, 2016 9th International Symposium on Computational Intelligence and Design (ISCID), pp. 338-340.

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.

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

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 *)

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).

Original entry on oeis.org

1, 1, 9, 457, 118961, 152894961, 940318147705, 26967408304580857, 3534888068831469959649, 2084993641133372935803249505, 5465706581663919414225671125834601, 63043356313898446097762231466174924913065, 3173076775252515207774429654590479617164788572049

Offset: 0

Seiichi Manyama, Jul 02 2025

nonn

Cf. A000272, A096538, A141369, A385526, A385528, A385529, A385530.

Cf. A015085.

nmax = 15; A[] = 1; Do[A[x] = E^(x*A[4*x]) + O[x]^j // Normal, {j, 1, nmax + 1}]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 02 2025 *)

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 A385527(n)
  A(4, n)
end

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

a(n) ~ c * n! * 2^(n*(n-1)), where c = 1.216702003338638031273833889488221691367428313263423339843... - Vaclav Kotesovec, Jul 02 2025

A348902 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(4*x)).

Original entry on oeis.org

1, 1, 9, 305, 39705, 20412737, 41846783913, 342892875489361, 11236600170415809849, 1472826135905484728387681, 772188014962631262957890704329, 1619397184353040716422147490531778929, 13584491414647344530078887450781292845554521

Offset: 0

Ilya Gutkovskiy, Nov 03 2021

nonn

Cf. A001003, A015085, A165941, A348188, A348901.

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

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

a(n) ~ c * 2^(n^2), where c = 2^(7/8) / EllipticTheta(2, 0, 1/sqrt(2)) = 0.6091497110662286155211146043057245512950999410185846745870491125003511... (same constant as in A165941). - Vaclav Kotesovec, Nov 03 2021, updated Apr 21 2024

A348859 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(4*x))).

Original entry on oeis.org

1, 2, 11, 204, 13701, 3550838, 3646912991, 14948746703872, 244965160945456921, 16054771878797715999594, 4208710286900635084866205491, 4413165224136772109314051383922356, 18510169791808150609141704979384516863021, 310549172324407121253872529077196811473762678750

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A007317, A015085, A348857, A348858.

nmax = 13; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[4 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[4^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]

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

a(n) ~ c * 2^(n*(n-1)), where c = 3.399782064170449155365557063612838469541502782488369640092639686931819... - Vaclav Kotesovec, Nov 02 2021

A348877 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(4*x)).

Original entry on oeis.org

1, 2, 12, 232, 15792, 4108192, 4223439552, 17316156716672, 283777228606348032, 18598759772257600748032, 4875627680189345535622228992, 5112485673116229482189477259405312, 21443339558695300334256395183459423465472, 359759625310995318218730673236935427042834358272

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A006318, A015085, A348875, A348876.

nmax = 13; A[] = 0; Do[A[x] = 1/(1 - x - x A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[4^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} 4^k * a(k) * a(n-k-1).

a(n) ~ c * 2^(n*(n-1)), where c = 2*Product_{j>=1} (4^j+1)/(4^j-1) = 3.938520707336538863894387393934531340132379924622409970534801850699757421... - Vaclav Kotesovec, Nov 03 2021

A348880 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(4*x)).

Original entry on oeis.org

1, 1, 2, 7, 45, 540, 12645, 578965, 52968266, 9592378291, 3490570329073, 2521575506955308, 3665174976025818601, 10583587128179171478201, 61512603105342112799632050, 710375545029057279438117199695, 16513584476995892580457952423234565

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A001006, A015085, A348878, A348879.

nmax = 16; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[4^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} 4^k * a(k) * a(n-k-2).

a(n) ~ c * 2^(n*(n-2)/2), where c = 3.18049189724646501466385558274654521200715578089919192312230814532162... - Vaclav Kotesovec, Nov 03 2021

A348862 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(4*x))).

Original entry on oeis.org

1, 0, 1, 16, 1041, 267552, 274242081, 1123570105392, 18409696460431921, 1206516278059945211200, 316282209730469497179053121, 331646250633753603369328903503952, 1391025527264722227030105092707830630481, 23337537123459992903665202300959789335795178848

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A005043, A015085, A348860, A348861.

nmax = 13; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[4 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + Sum[4^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]

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

a(n) ~ c * 2^(n*(n-1)), where c = 0.2554910592341818819974992745952574870516320592891123415106817713508566833... - Vaclav Kotesovec, Nov 02 2021

A352008 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, 5, 9, 29, 65, 437, 953, 3981, 10097, 49829, 123241, 516349, 1400737, 10203285, 24698905, 111642477, 304787665, 1704790917, 4392726473, 19951366877, 56296655617, 336083829621, 878995865721, 3974885167949, 11362790432305, 60789762148453, 165051865924137

Offset: 0

Ilya Gutkovskiy, Feb 28 2022

nonn

Cf. A000621, A015085, A352006, A352007.

a[0] = 1; a[n_] := a[n] = Sum[4^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1/(1 - x A[4 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

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

A376097 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^4 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 17, 1410, 364019, 228282823, 296324235500, 712075198644414, 2918094100584013255, 19151474626728425949663, 191553141880332262049655201, 2804913258838830873001491036584, 58168297154586087400230338311689652, 1661461159115675581245556180230933084340

Offset: 0

Ilya Gutkovskiy, Sep 10 2024

nonn

Cf. A000699, A015085, A088716, A256020, A376095, A376096.

a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^4 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 15 x^2 A[x] A'[x] + 25 x^3 A[x] A''[x] + 10 x^4 A[x] A'''[x] + x^5 A[x] A''''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 15 * x^2 * A(x) * A'(x) + 25 * x^3 * A(x) * A''(x) + 10 * x^4 * A(x) * A'''(x) + x^5 * A(x) * A''''(x).

cp's OEIS Frontend

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 - ...)))))).

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

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A385527 E.g.f. A(x) satisfies A(x) = exp(x*A(4*x)).

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A348902 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(4*x)).

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A348859 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(4*x))).

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A348877 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(4*x)).

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A348880 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(4*x)).

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A348862 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(4*x))).

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A352008 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * a(k) * a(n-2*k-1).

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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 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).

A385527 E.g.f. A(x) satisfies A(x) = exp(xA(4x)).