A099235 -id:A099235 - cp's OEIS frontend

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 8, 1, 1, 1, 6, 15, 26, 28, 13, 1, 1, 1, 7, 21, 45, 69, 60, 21, 1, 1, 1, 8, 28, 71, 140, 181, 129, 34, 1, 1, 1, 9, 36, 105, 251, 431, 476, 277, 55, 1, 1, 1, 10, 45, 148, 413, 882, 1326, 1252, 595, 89, 1

Offset: 0

Paul Barry, Oct 08 2004

			Rows begin
  1, 1, 1,  1,  1,   1, ...
  1, 1, 2,  3,  5,   8, ...
  1, 1, 3,  6, 13,  28, ...
  1, 1, 4, 10, 26,  69, ...
  1, 1, 5, 15, 45, 140, ...
Row 1 is the 0-section of 1/(1-x-x)   (A000079);
Row 2 is the 1-section of 1/(1-x-x^2) (A000045);
Row 3 is the 2-section of 1/(1-x-x^3) (A000930);
Row 4 is the 3-section of 1/(1-x-x^4) (A003269);
etc.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Sums of antidiagonals are A099236.
Columns include A000217, A008778.
Rows include A000045, A002478, A099234, A099235.
Main diagonal gives A099237.
Cf. A099238.

Square array T(n, k) = Sum_{j=0..n} binomial(k(n-j), j).

Rows are generated by 1/(1-x(1+x)^k) and satisfy a(n) = Sum_{k=0..n} binomial(n, k)a(n-k-1).

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).

Original entry on oeis.org

1, 1, 6, 27, 134, 709, 3892, 22004, 127250, 749230, 4476386, 27071344, 165398868, 1019405720, 6330482488, 39571612357, 248796862550, 1572300095758, 9981970108384, 63633339713190, 407162295120570, 2614059813642256, 16834457481559076

Offset: 0

Seiichi Manyama, Jan 25 2023

nonn

Cf. A052709, A073155, A360076, A360083.

Cf. A000108, A099235.

a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(2*k, k)/(k+1));

my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^4)))

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

G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^4 )).

A360088 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*k,n-k).

Original entry on oeis.org

1, -1, -3, 1, 13, 4, -49, -46, 165, 284, -476, -1417, 1003, 6220, -110, -24644, -14831, 88184, 113224, -278288, -619744, 715647, 2891977, -1036173, -12068353, -3381661, 45588556, 41600921, -154355594, -259984429, 448828716, 1305250324, -964837159, -5754843123

Offset: 0

Seiichi Manyama, Jan 25 2023

Index entries for linear recurrences with constant coefficients, signature (-1,-4,-6,-4,-1).

Cf. A077979, A360087, A360089.

Cf. A099235.

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

my(N=40, x='x+O('x^N)); Vec(1/(1+x*(1+x)^4))

a(n) = -a(n-1) - 4*a(n-2) - 6*a(n-3) - 4*a(n-4) - a(n-5).

G.f.: 1/(1 + x*(1+x)^4).

A360090 a(n) = Sum_{k=0..n} binomial(5*k,n-k).

Original entry on oeis.org

1, 1, 6, 21, 71, 251, 882, 3088, 10829, 37975, 133146, 466852, 1636944, 5739647, 20125051, 70564951, 247423522, 867546829, 3041899638, 10665883415, 37398034921, 131129599227, 459782762029, 1612146986543, 5652708454881, 19820223058176, 69496108849357

Offset: 0

Seiichi Manyama, Jan 25 2023

The number of ways to place non-overlapping Young diagrams of shape (2,1,1,1,1) on an 9 by n rectangle. - Per Alexandersson, Jul 01 2025

Index entries for linear recurrences with constant coefficients, signature (1,5,10,10,5,1).

Cf. A002478, A099234, A099235.

seq(add(binomial(5*k,n-k),k=0..n), n=0..50); # Robert Israel, Jul 09 2025

a(n) = sum(k=0, n, binomial(5*k, n-k));

my(N=30, x='x+O('x^N)); Vec(1/(1-x*(1+x)^5))

a(n) = a(n-1) + 5*a(n-2) + 10*a(n-3) + 10*a(n-4) + 5*a(n-5) + a(n-6).

G.f.: 1/(1 - x*(1+x)^5).

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

Original entry on oeis.org

1, 1, 2, 12, 60, 460, 3900, 39438, 456288, 5896224, 85230000, 1349017560, 23353941600, 437432418696, 8828284404576, 190867622500800, 4401749312069760, 107859517575659520, 2798352667710645120, 76636669899079699776, 2209235394261812751360

Offset: 0

Seiichi Manyama, Apr 12 2023

nonn

Cf. A202152, A362237.

Cf. A002478, A099234, A099235.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*(1+x)^x)))

a(n) = n! * Sum_{i=0..n} Sum_{j=0..n-i} i^j * Stirling1(n-i-j,j)/(n-i-j)!.

A366222 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^3.

Original entry on oeis.org

1, 1, 7, 42, 287, 2114, 16338, 130802, 1075355, 9025656, 77021482, 666267502, 5829209046, 51492030953, 458612500526, 4113879873624, 37133888342707, 337041718357465, 3074153880004188, 28162578841220534, 259020296989987934, 2390818256963083305

Offset: 0

Seiichi Manyama, Oct 04 2023

nonn

Cf. A364475, A366200, A366221.

Cf. A099235, A360082.

a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..n} binomial(4*k,n-k) * binomial(3*k,k)/(2*k+1).

cp's OEIS Frontend

A099233 Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

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A360082 a(n) = Sum_{k=0..n} binomial(4*k,n-k) * Catalan(k).

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PARI

PARI

Formula

A360088 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*k,n-k).

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PARI

PARI

Formula

A360090 a(n) = Sum_{k=0..n} binomial(5*k,n-k).

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Maple

PARI

PARI

Formula

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

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Programs

PARI

Formula

A366222 G.f. A(x) satisfies A(x) = 1 + x*(1 + x)^4*A(x)^3.

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Author

Keywords

Crossrefs

Programs

PARI

Formula

A360082 a(n) = Sum_{k=0..n} binomial(4k,n-k) Catalan(k).

A366222 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^4A(x)^3.