A328747 -id:A328747 - cp's OEIS frontend

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11

Offset: 0

Seiichi Manyama, Jul 06 2019

A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019
Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below).  Then A(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

			Square array, A(n, k), begins:
   1,  1,   1,    1,     1,      1, ... A000012;
   2,  2,   2,    2,     2,      2, ... A007395;
   3,  4,   6,   10,    18,     34, ... A052548;
   4,  8,  20,   56,   164,    488, ... A115099;
   5, 16,  70,  346,  1810,   9826, ...
   6, 32, 252, 2252, 21252, 206252, ...
Antidiagonals, T(n, k), begin:
  1;
  1,  2;
  1,  2,   3;
  1,  2,   4,    4;
  1,  2,   6,    8,    5;
  1,  2,  10,   20,   16,     6;
  1,  2,  18,   56,   70,    32,     7;
  1,  2,  34,  164,  346,   252,    64,    8;
  1,  2,  66,  488, 1810,  2252,   924,  128,   9;
  1,  2, 130, 1460, 9826, 21252, 15184, 3432, 256,  10;

R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Main diagonal gives A167010.
Cf. A328747, A328748, A328807, A328812.
Cf. A000012, A007395, A052548, A115099.

[(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022

nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)

A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022

flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 26 2022

A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
A(n, n+1) = A328812(n).
A(n, n) = A167010(n).
T(n, k) = A(k, n-k) (antidiagonals).
T(n, n) = A000027(n+1).
T(n, n-1) = A000079(n-1).
T(n, n-2) = A000984(n-2).
T(n, n-3) = A000172(n-3).
T(n, n-4) = A005260(n-4).
T(n, n-5) = A005261(n-5).
T(n, n-6) = A069865(n-6).
T(n, n-7) = A182421(n-7).
T(n, n-8) = A182422(n-8).
T(n, n-9) = A182446(n-9).
T(n, n-10) = A182447(n-10).
T(n, n-11) = A342294(n-11).
T(n, n-12) = A342295(n-12).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

Original entry on oeis.org

1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Inverse binomial transform of the Franel numbers (A000172). - Paul D. Hanna, Feb 26 2012
a(n) is the constant term in the expansion of (1 + x + y + 1/x + 1/y + x/y + y/x)^n. - Seiichi Manyama, Oct 26 2019
a(n) is the constant term in the expansion of (-1 + (1 + x) * (1 + y) + (1 + 1/x) * (1 + 1/y))^n. - Seiichi Manyama, Oct 27 2019
a(n) is the number of n step closed walks on the hexagonal lattice with loops at each node. A step along a loop leaves the position unchanged. The bijection is as follows: after subtracting 1 from each element in the array, values are -1, 0 or 1 and row and column sums are zero. There are only seven possibilities for each row. An all zero row corresponds with a step along the loop leaving the position unchanged and the others to a unit step in each of the six possible directions. This justifies that this sequence is the binomial transform of A002898. - Andrew Howroyd, May 09 2020

			G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...
G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...

Andrew Howroyd, Table of n, a(n) for n = 0..500 (terms 1..99 from R. H. Hardin)

Column k=3 of A328747 and A334549.

Cf. A000172, A002898, A006480, A328725.

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1),{k,0,n}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(sum(m=0,n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x + x*O(x^n))^(3*m+1)),n)} \\ Paul D. Hanna, Feb 26 2012

a(n)={sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3))} \\ Andrew Howroyd, May 09 2020

From Paul D. Hanna, Feb 26 2012: (Start)
G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).
Equals the binomial transform of A002898.
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.
(End)
Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012
G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

a(0)=1 prepended by Andrew Howroyd, May 09 2020

A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, 0, 2, 1, 0, 2, 0, -3, 1, 0, 6, 0, 0, 4, 1, 0, 14, 12, 6, 0, -5, 1, 0, 30, 72, 90, 0, 0, 6, 1, 0, 62, 300, 882, 360, 20, 0, -7, 1, 0, 126, 1080, 6690, 8400, 2040, 0, 0, 8, 1, 0, 254, 3612, 44706, 124920, 95180, 10080, 70, 0, -9

Offset: 0

Seiichi Manyama, Oct 27 2019

T(n,k) is the constant term in the expansion of (-2 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

			Square array begins:
    1, 1, 1,   1,    1,      1, ...
    0, 0, 0,   0,    0,      0, ...
   -1, 0, 2,   6,   14,     30, ...
    2, 0, 0,  12,   72,    300, ...
   -3, 0, 6,  90,  882,   6690, ...
    4, 0, 0, 360, 8400, 124920, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..5 give A097141(n+1), A000007, A126869, A002898, A328735, A328751.
T(n,n+1) gives A328814.
Cf. A309010, A328747, A328807.

T[n_, k_] := Sum[(-2)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 9, 20, 1, 3, 11, 27, 48, 1, 3, 15, 45, 81, 112, 1, 3, 23, 93, 195, 243, 256, 1, 3, 39, 225, 639, 873, 729, 576, 1, 3, 71, 597, 2583, 4653, 3989, 2187, 1280, 1, 3, 135, 1665, 11991, 32133, 35169, 18483, 6561, 2816

Offset: 0

Seiichi Manyama, Oct 28 2019

T(n,k) is the constant term in the expansion of (1 + Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0.

For fixed k > 0 is T(n,k) ~ (2^k + 1)^(n + (k-1)/2) / (2^((k-1)^2/2) * sqrt(k) * (Pi*n)^((k-1)/2)). - Vaclav Kotesovec, Oct 28 2019

			Square array begins:
     1,   1,   1,    1,     1,      1, ...
     3,   3,   3,    3,     3,      3, ...
     8,   9,  11,   15,    23,     39, ...
    20,  27,  45,   93,   225,    597, ...
    48,  81, 195,  639,  2583,  11991, ...
   112, 243, 873, 4653, 32133, 260613, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..5 give A001792, A000244, A026375, A002893, A328808, A328809.
Main diagonal gives A328810.
Cf. A309010, A328747, A328748.

T[n_, k_] := Sum[Binomial[n, i] * Sum[Binomial[i, j]^k, {j, 0, i}], {i, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

A328750 Constant term in the expansion of (-1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

Original entry on oeis.org

1, 1, 31, 391, 8071, 161671, 3634921, 84109201, 2032357111, 50355327991, 1277302604521, 32983865502721, 864982811998801, 22976755021842961, 617140285389771391, 16735405610179740151, 457647302453165769751, 12607719926638032161431, 349620344754345216824041

Offset: 0

Seiichi Manyama, Oct 27 2019

nonn

Column k=5 of A328747.

Cf. A005261, A328751.

Table[Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[i, j]^5, {j, 0, i}], {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n,i)*sum(j=0, i, binomial(i, j)^5))}

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^5.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^4*(440*n^2 - 2728*n + 3723)*a(n) = (6600*n^6 - 54120*n^5 + 147013*n^4 - 174348*n^3 + 102442*n^2 - 29260*n + 3108)*a(n-1) + (194920*n^6 - 1988184*n^5 + 7650713*n^4 - 14588908*n^3 + 14793198*n^2 - 7658420*n + 1601964)*a(n-2) + (n-2)*(690800*n^5 - 7046160*n^4 + 26712814*n^3 - 47822370*n^2 + 40779795*n - 13361628)*a(n-3) + (n-3)*(n-2)*(975480*n^4 - 8974416*n^3 + 28602923*n^2 - 37477643*n + 16905924)*a(n-4) + (n-4)*(n-3)*(n-2)*(622600*n^3 - 4482720*n^2 + 9455173*n - 5628497)*a(n-5) + 341*(n-5)*(n-4)*(n-3)*(n-2)*(440*n^2 - 1848*n + 1435)*a(n-6).
a(n) ~ 31^(n+2) / (256 * sqrt(5) * Pi^2 * n^2). (End)

A328813 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

Original entry on oeis.org

1, 1, 7, 115, 8071, 1770951, 1505946121, 4368457532265, 49949721645153751, 2021436054924485283799, 327902645022367779788597977, 191573267131797606250658812550565, 453516825886934673673734108656254582801

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A328747, A328812, A328814.

a[n_] := Sum[(-1)^(n-i) * Binomial[n, i] * Sum[Binomial[i, j]^(n+1), {j, 0, i}], {i, 0, n}]; Array[a, 13, 0] (* Amiram Eldar, May 06 2021 *)

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^(n+1)))}

a(n) = A328747(n,n+1) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^(n+1).

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^n.

Original entry on oeis.org

1, 1, 3, 31, 1255, 161671, 75481581, 121338954577, 734884394666535, 15970479086714049751, 1347242827078365957146473, 415839472158527880691583531617, 507266883682599825619985300960971525, 2284735689605775548174387143718048664963601

Offset: 0

Seiichi Manyama, Oct 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Main diagonal of A328747.

Cf. A167010, A328810.

Table[Sum[(-1)^(n-i)*Binomial[n, i]*Sum[Binomial[i, j]^n, {j, 0, i}], {i, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 28 2019 *)

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^n))}

a(n) ~ A167010(n). - Vaclav Kotesovec, Oct 28 2019

cp's OEIS Frontend

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

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A172634 Number of n X 3 0..2 arrays with row sums 3 and column sums n.

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A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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A328807 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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A328750 Constant term in the expansion of (-1 + (1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n.

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A328813 Constant term in the expansion of (-1 + Product_{k=1..n} (1 + x_k) + Product_{k=1..n} (1 + 1/x_k))^n.

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A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^n.

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A328748 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-2)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

A328811 a(n) = Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^n.