A099233 -id:A099233 - cp's OEIS frontend

A099236 Sums of antidiagonals of A099233.

1, 2, 3, 5, 9, 18, 39, 91, 226, 594, 1643, 4763, 14419, 45444, 148714, 504150, 1766930, 6390720, 23815168, 91306968, 359694457, 1454213026, 6027213531, 25583995337, 111118605583, 493407322280, 2238131105770, 10363617299670, 48954143141361

Offset: 0

Paul Barry, Oct 08 2004

a(n)=sum{j=0..n, sum{k=0..n-j, binomial(j(n-j-k), k)}}

A099234 A trisection of 1/(1-x-x^4).

Original entry on oeis.org

1, 1, 4, 10, 26, 69, 181, 476, 1252, 3292, 8657, 22765, 59864, 157422, 413966, 1088589, 2862617, 7527704, 19795288, 52054840, 136886433, 359964521, 946583628, 2489191330, 6545722210, 17213011605, 45264335853, 119029728628

Offset: 0

Paul Barry, Oct 08 2004

A row of A099233.

Row sums of number triangle A116089. - Paul Barry, Feb 04 2006

Michael De Vlieger, Table of n, a(n) for n = 0..2382
Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, Schreier Multisets and the s-step Fibonacci Sequences, arXiv:2304.05409 [math.CO], 2023. See also Integers (2024) Vol. 24A, Art. No. A7, p. 3.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (1,3,3,1).

Cf. A003269, A099233, A116089.

CoefficientList[Series[1/(1-x (1+x)^3),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,3,1},{1,1,4,10},30] (* Harvey P. Dale, Jun 05 2011 *)

G.f.: 1/(1-x*(1+x)^3).
a(n) = Sum_{k=0..n} binomial(3*(n-k), k).
a(n) = a(n-1)+3*a(n-2)+3*a(n-3)+a(n-4).
a(n) = A003269(3n).
a(n) = Sum_{k=0..n} C(3*k,n-k) = Sum_{k=0..n} C(n,k)*C(4*k,n)/C(4*k,k). - Paul Barry, Feb 04 2006
G.f.: 1/(G(0) - x) where G(k) = 1 - (2*k+3)*x/(2*k+1 - x*(k+2)*(2*k+1)/(x*(k+2) - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2012

A099237 a(n) = Sum_{k=0..n} binomial(n*(n-k), k).

Original entry on oeis.org

1, 1, 3, 10, 45, 251, 1624, 11908, 97545, 880660, 8664546, 92096731, 1050304775, 12778138842, 165033693175, 2253204163256, 32401745953105, 489207829112931, 7733130368443057, 127664099576228184, 2196149923000824756

Offset: 0

Paul Barry, Oct 08 2004

Main diagonal of A099233.

Vaclav Kotesovec, Table of n, a(n) for n = 0..490

Cf. A099233, A157114, A359842.

A099237:= func< n | (&+[Binomial(n*j, n-j): j in [0..n]]) >;
[A099237(n): n in [0..30]]; // G. C. Greubel, Mar 09 2021

A099237:= n-> add( binomial(n*j, n-j), j=0..n );
seq(A099237(n), n=0..30); # G. C. Greubel, Mar 09 2021

Table[Sum[Binomial[n*(n - k), k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Feb 19 2018 *)

def A099237(n): return sum( binomial(n*j, n-j) for j in (0..n))
[A099237(n) for n in (0..30)] # G. C. Greubel, Mar 09 2021

From Vaclav Kotesovec, Feb 19 2018: (Start)
a(n)^(1/n) ~ n^(n/w) * (n+1-w)^(1 - (n+1)/w) * (w-1)^(1/w - 1), where w = LambertW(exp(1)*n),
a(n)^(1/n) ~ n/log(n), but the convergence is too slow. (End)
From Peter Bala, Jan 19 2023: (Start)
Conjectures: a(2^k) == 1 (mod 2^k) and a(3^k) == 1 (mod 3^(k+1)); a(p^k) == 1 (mod p^(k+1)) for all primes p >= 5.
Let m be a positive integer. Similar recurrences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(m*n*k, n-k). Cf. A359842. (End)

A099235 A quadrisection of 1/(1-x-x^5).

Original entry on oeis.org

1, 1, 5, 15, 45, 140, 431, 1326, 4085, 12580, 38740, 119305, 367411, 1131476, 3484490, 10730820, 33046585, 101770120, 313410816, 965178576, 2972359720, 9153665985, 28189589705, 86812537085, 267347509271, 823322219501

Offset: 0

Paul Barry, Oct 08 2004

A row of A099233.

The number of ways to place non-overlapping Young diagrams of shape (2,1,1,1) on an 7 by n rectangle. - Per Alexandersson, Jun 23 2025

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,6,4,1).

Cf. A003520, A099233.

Take[CoefficientList[Series[1/(1-x-x^5),{x,0,100}],x],{1,-1,4}] (* or *) LinearRecurrence[{1,4,6,4,1},{1,1,5,15,45},30] (* Harvey P. Dale, Mar 06 2015 *)

G.f.: 1/(1-x*(1+x)^4).
a(n) = Sum_{k=0..n} binomial(4(n-k), k).
a(n) = a(n-1) + 4*a(n-2) + 6*a(n-3) + 4*a(n-4) + a(n-5).
a(n) = A003520(4n).

A099238 Square array read by antidiagonals with rows generated by 1/(1-x-x^(k+1)).

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 1, 2, 8, 1, 1, 1, 3, 16, 1, 1, 1, 2, 5, 32, 1, 1, 1, 1, 3, 8, 64, 1, 1, 1, 1, 2, 4, 13, 128, 1, 1, 1, 1, 1, 3, 6, 21, 256, 1, 1, 1, 1, 1, 2, 4, 9, 34, 512, 1, 1, 1, 1, 1, 1, 3, 5, 13, 55, 1024, 1, 1, 1, 1, 1, 1, 2, 4, 7, 19, 89, 2048

Offset: 0

Paul Barry, Oct 08 2004

Sections of rows are given by array A099233. Sums of antidiagonals yield A097939.

The triangle of diagonals terminated after reaching the repeating value is A329146. - Andrey Zabolotskiy, Sep 01 2020

			Rows begin
1,   2,   4,   8,  16,  32,  64, 128, 256, ... (A000079)
1,   1,   2,   3,   5,   8,  13,  21,  34, ... (A000045)
1,   1,   1,   2,   3,   4,   6,   9,  13, ... (A000930)
1,   1,   1,   1,   2,   3,   4,   5,   7, ... (A003269)
1,   1,   1,   1,   1,   2,   3,   4,   5, ... (A003520)

Square array T(n, k) = Sum_{j=0..floor(n/(k+1))} binomial(n-k*j, j), n, k>=0.

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    2,   2,    2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   46,   62,   80,  100, ...
   70, 136,  226,  342,  486,  660, ...
  252, 592, 1136, 1932, 3030, 4482, ...

Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Cf. A099233, A361834.

T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));

G.f. of column k: 1/sqrt(1 - 4*x*(1+x)^k).

n*T(n,k) = 2 * Sum_{j=0..k} binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A373717 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(kn/(2k+1))} binomial(k * (n-2*j),j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 5, 7, 8, 6, 1, 1, 1, 1, 6, 9, 13, 15, 9, 1, 1, 1, 1, 7, 11, 19, 28, 26, 13, 1, 1, 1, 1, 8, 13, 26, 45, 53, 45, 19, 1, 1, 1, 1, 9, 15, 34, 66, 91, 105, 80, 28, 1, 1, 1, 1, 10, 17, 43, 91, 141, 201, 211, 140, 41, 1

Offset: 0

Seiichi Manyama, Jun 15 2024

			Square array begins:
  1, 1,  1,  1,  1,  1,  1, ...
  1, 1,  1,  1,  1,  1,  1, ...
  1, 1,  1,  1,  1,  1,  1, ...
  1, 2,  3,  4,  5,  6,  7, ...
  1, 3,  5,  7,  9, 11, 13, ...
  1, 4,  8, 13, 19, 26, 34, ...
  1, 6, 15, 28, 45, 66, 91, ...

Columns k=0..3 give A000012, A000930, A193147, A373718.
Main diagonal gives A373719.
Cf. A099233.

T(n, k) = sum(j=0, k*n\(2*k+1), binomial(k*(n-2*j), j));

G.f. of column k: 1/(1 - x * (1 + x^2)^k).

T(n,k) = Sum_{j=0..k} binomial(k,j) * T(n-2*j-1,k).

cp's OEIS Frontend

A099236 Sums of antidiagonals of A099233.

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A099234 A trisection of 1/(1-x-x^4).

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A099237 a(n) = Sum_{k=0..n} binomial(n*(n-k), k).

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A099235 A quadrisection of 1/(1-x-x^5).

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A099238 Square array read by antidiagonals with rows generated by 1/(1-x-x^(k+1)).

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A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).

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A373717 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(k*n/(2*k+1))} binomial(k * (n-2*j),j).

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A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

A373717 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..floor(kn/(2k+1))} binomial(k * (n-2*j),j).