A362079 -id:A362079 - cp's OEIS frontend

A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.

1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 22, 35, 1, 1, 9, 37, 105, 126, 1, 1, 11, 55, 215, 511, 462, 1, 1, 13, 76, 369, 1271, 2534, 1716, 1, 1, 15, 100, 571, 2526, 7651, 12720, 6435, 1, 1, 17, 127, 825, 4401, 17577, 46614, 64449, 24310, 1, 1, 19, 157, 1135, 7026, 34412, 123810, 286599, 328900, 92378

Offset: 0

Seiichi Manyama, Apr 08 2023

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    1,   1,    1,    1,    1,    1, ...
    3,   5,    7,    9,   11,   13, ...
   10,  22,   37,   55,   76,  100, ...
   35, 105,  215,  369,  571,  825, ...
  126, 511, 1271, 2526, 4401, 7026, ...

Columns k=0..3 give A088218, A213684, A362087, A362088.
Main diagonal gives A362080.
Cf. A362079.

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

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

A362080 a(n) = [x^n] 1/(1 - x*(1+x)^n)^n.

Original entry on oeis.org

1, 1, 7, 55, 571, 7026, 98925, 1562219, 27214867, 516646333, 10589130223, 232574622116, 5440521381816, 134859301929873, 3527034780915985, 96965997588549555, 2793286163779275779, 84076751617833902070, 2637677096916448507104

Offset: 0

Seiichi Manyama, Apr 08 2023

Seiichi Manyama, Table of n, a(n) for n = 0..449

Main diagonal of A362078.
Main diagonal of A362079.
Cf. A099237.

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

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

A362084 a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 7, 28, 145, 896, 6328, 50212, 441489, 4248370, 44306306, 496991848, 5959111223, 75977511442, 1025441134747, 14594189335496, 218290221112929, 3421314388169988, 56043004143343843, 957209642080023286, 17011439135301438016, 313980693855333453740

Offset: 0

Seiichi Manyama, Apr 08 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..493

Column k=2 of A362079.

Cf. A099237.

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

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

a(n) = [x^n] 1/(1 - x*(1+x)^n)^2.

A362085 a(n) = Sum_{k=0..n} (-1)^k * binomial(-3,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 3, 12, 55, 315, 2106, 15946, 134730, 1253637, 12702961, 138955146, 1629581955, 20371061009, 270124999977, 3783651174906, 55780472480036, 862795543656489, 13963065117796548, 235845816764772718, 4148499020022749151, 75841424406989195136

Offset: 0

Seiichi Manyama, Apr 08 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..492

Column k=3 of A362079.

Cf. A000217, A362088.

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

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

a(n) = [x^n] 1/(1 - x*(1+x)^n)^3.

A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 18, 5, 0, 1, 5, 26, 55, 47, 8, 0, 1, 6, 40, 124, 198, 118, 13, 0, 1, 7, 57, 235, 571, 681, 290, 21, 0, 1, 8, 77, 398, 1320, 2500, 2263, 702, 34, 0, 1, 9, 100, 623, 2640, 7026, 10504, 7341, 1677, 55, 0

Offset: 0

Seiichi Manyama, Apr 08 2023

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   7,  15,   26,   40, ...
  0, 3,  18,  55,  124,  235, ...
  0, 5,  47, 198,  571, 1320, ...
  0, 8, 118, 681, 2500, 7026, ...

Columns k=0..3 give A000007, A000045(n+1), A362126, A382614.
Main diagonal gives A362080.
Cf. A362078, A362079.

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

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

cp's OEIS Frontend

A362078 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = [x^n] 1/(1 - x*(1+x)^k)^n.

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A362080 a(n) = [x^n] 1/(1 - x*(1+x)^n)^n.

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A362084 a(n) = Sum_{k=0..n} (-1)^k * binomial(-2,k) * binomial(n*k,n-k).

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A362085 a(n) = Sum_{k=0..n} (-1)^k * binomial(-3,k) * binomial(n*k,n-k).

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A362125 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - x*(1+x)^k)^k.

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