A172634 -id:A172634 - cp's OEIS frontend

A208446 Duplicate of A172634.

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

Offset: 0

dead

Table of n, a(n) for n = 0..20

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1

Offset: 0

Andrew Howroyd, May 09 2020

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.

			Array begins:
====================================================================
n\k | 0 1   2     3       4          5            6            7
----|---------------------------------------------------------------
  0 | 1 1   1     1       1          1            1            1 ...
  1 | 1 1   1     1       1          1            1            1 ...
  2 | 1 1   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]

Andrew Howroyd, Table of n, a(n) for n = 0..350 (first 26 antidiagonals)

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
Main diagonal is A172645.
Cf. A008300, A333901, A376935, A377063 (up to row permutations).

T(n,k) = T(k,n).

A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 7, 1, 0, 1, 1, 15, 31, 19, 1, 0, 1, 1, 31, 115, 175, 51, 1, 0, 1, 1, 63, 391, 1255, 991, 141, 1, 0, 1, 1, 127, 1267, 8071, 13671, 5881, 393, 1, 0, 1, 1, 255, 3991, 49399, 161671, 160461, 35617, 1107, 1, 0

Offset: 0

Seiichi Manyama, Oct 27 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, 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, ...
   1, 1,  1,   1,     1,      1, ...
   0, 1,  3,   7,    15,     31, ...
   0, 1,  7,  31,   115,    391, ...
   0, 1, 19, 175,  1255,   8071, ...
   0, 1, 51, 991, 13671, 161671, ...

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

Columns k=0..5 give A019590(n+1), A000012, A002426, A172634, A328725, A328750.
Main diagonal gives A328811.
T(n,n+1) gives A328813.
Cf. A309010, A328748, A328807.

T[n_, k_] := Sum[(-1)^(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 *)

A328725 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 1, 15, 115, 1255, 13671, 160461, 1936425, 24071895, 305313415, 3939158905, 51521082405, 681635916325, 9105864515125, 122657982366375, 1664151758259915, 22720725637684215, 311933068664333175, 4303704125389134825, 59640225721889127525, 829774531966386480705

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..854

Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A002426 (m=2), A172634 (m=3), this sequence (m=4), A328750 (m=5).

Cf. A002899, A005260, A328713.

{a(n) = polcoef(polcoef(polcoef((-1+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, (-1)^(n-i)*binomial(n,i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Oct 28 2019: (Start)
Recurrence: n^3*a(n) = (2*n - 1)^3*a(n-1) + (n-1)*(94*n^2 - 188*n + 93)*a(n-2) + 80*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 75*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 15^(n + 3/2) / (2^(11/2) * Pi^(3/2) * n^(3/2)). (End)

A262704 Triangle: Newton expansion of C(n,m)^3, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 6, 1, 0, 6, 24, 1, 0, 0, 114, 60, 1, 0, 0, 180, 690, 120, 1, 0, 0, 90, 2940, 2640, 210, 1, 0, 0, 0, 5670, 21840, 7770, 336, 1, 0, 0, 0, 5040, 87570, 107520, 19236, 504, 1, 0, 0, 0, 1680, 189000, 735210, 407400, 42084, 720, 1, 0, 0, 0, 0, 224700, 2835756, 4280850, 1284360, 83880, 990, 1

Offset: 0

Giuliano Cabrele, Sep 27 2015

Triangle here T_3(n,m) is such that C(n,m)^3 = Sum_{j=0..n} C(n,j)*T_3(j,m).
Equivalently, lower triangular matrix T_3 such that
|| C(n,m)^3 || = A181583 =  P * T_3 = A007318 * T_3.
T_3(n,m) = 0 for n < m and for 3*m < n. In fact:
C(x,m)^q and C(x,m), with m nonnegative and q positive integer, are polynomial in x of degree m*q and m respectively, and C(x,m) is a divisor of C(x,m)^q.
Therefore the Newton series will give C(x,m)^q = T_q(m,m)*C(x,m) + T_q(m+1,m)*C(x,m+1) + ... + T_q(q*m,m)*C(x,q*m), where T_q(n,m) is the n-th forward finite difference of C(x,m)^q at x = 0.
Example:
C(x,2)^3 = x^3*(x-1)^3 / 8 = 1*C(x,2) + 24*C(x,3) + 114*C(x,4) + 180*C(x,5) + 90*C(x,6);
C(5,2)^3 = C(5,3)^3 = 1000 = 1*C(5,2) + 24*C(5,3) + 114*C(5,4) + 180*C(5,5) = 1*C(5,3) + 60*C(5,4) + 690*C(5,5).
So we get the expansion of the 3rd power of the binomial coefficient in terms of the binomial coefficients on the same row.
T_1 is the unitary matrix,
T_2 is the transpose of A109983,
T_3 is this sequence,
T_4, T_5 are A262705, A262706.

			Triangle starts:
n\m  [0]     [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[0]  1;
[1]  0,      1;
[2]  0,      6,      1;
[3]  0,      6,      24,     1;
[4]  0,      0,      114,    60,     1;
[5]  0,      0,      180,    690,    120,    1;
[6]  0,      0,      90,     2940,   2640,   210,    1;
[7]  0,      0,      0,      5670,   21840,  7770,   336,    1;
[8]  0,      0,      0,      5040,   87570,  107520, 19236,  504,    1;
[9]  ...

Gheorghe Coserea, Rows n = 0..200, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

Row sums are A172634, the inverse binomial transform of the Franel numbers (A000172).
Column sums are the A126086, per the comment given thereto by Brendan McKay.
Second diagonal (T_3(n+1,n)) is A007531 (n+2).
Column T_3(n,2) is A122193(3,n).
Cf. A109983 (transpose of), A262705, A262706.
Cf. A078739, A078741, A274786.

[&+[(-1)^(n-j)*Binomial(n,j)*Binomial(j,m)^3: j in [0..n]]: m in [0..n], n in [0..10]]; // Bruno Berselli, Oct 01 2015

T3[n_, m_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[j, m]^3, {j, 0, n}]; Table[T3[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Oct 01 2015 *)

// as a function
T_3:=(n,m)->_plus((-1)^(n-j)*binomial(n,j)*binomial(j,m)^3 $ j=0..n):
// as a matrix h x h
_P:=h->matrix([[binomial(n,m) $m=0..h]$n=0..h]):
_P_3:=h->matrix([[binomial(n,m)^3 $m=0..h]$n=0..h]):
_T_3:=h->_P(h)^-1*_P_3(h):

T_3(nmax) = {for(n=0, nmax, for(m=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n,j)*binomial(j,m)^3), ", ")); print())} \\ Colin Barker, Oct 01 2015

t3(n,m) = sum(j=0, n,  (-1)^((n-j)%2)* binomial(n,j)*binomial(j,m)^3);
concat(vector(11, n, vector(n, k, t3(n-1,k-1)))) \\ Gheorghe Coserea, Jul 14 2016

T_3(n,m) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(j,m)^3.

Also, let S(r,s)(n,m) denote the Generalized Stirling2 numbers as defined in the link above,then T_3(n,m) = n! / (m!)^3 * S(m,m)(3,n).

A377065 Number of 3 X n 0..2 matrices with row sums n and column sums 3 up to permutations of rows.

Original entry on oeis.org

1, 1, 2, 6, 30, 166, 981, 5937, 36646, 229350, 1451757, 9274057, 59699729, 386798777, 2520034050, 16497343046, 108454221206, 715629888822, 4737625385061, 31456633327905, 209418369288865, 1397521222483385, 9346484009527370, 62632803958053870, 420481623373564025

Offset: 0

Andrew Howroyd, Oct 15 2024

nonn

Also, the number of 3 X n {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.

			The a(2) = 2 matrices are:
  [1 1]  [2 0]
  [1 1]  [0 2]
  [1 1]  [1 1]
The a(3) = 6 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]  [2 1 0]  [2 0 1]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]  [1 0 2]  [1 2 0]
  [1 1 1]  [1 1 1]  [1 1 1]  [1 1 1]  [0 2 1]  [0 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..500

Row n=3 of A377063.

Cf. A172634, A377066, A377067.

a(n)={(5+sum(i=0, n, sum(j=0, i, (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)))/6}

a(n) = (A172634(n) - 1)/6 + 1.

a(n) = (5 + Sum_{i=0..n} Sum_{j=0..i} (-1)^(n-i)*binomial(n, i)*binomial(i, j)^3)/6.

A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.

Original entry on oeis.org

1, 1, 4, 6, 12, 18, 30, 42, 63, 85, 118, 154, 204, 258, 330, 408, 507, 615, 748, 892, 1066, 1254, 1476, 1716, 1995, 2295, 2640, 3010, 3430, 3880, 4386, 4926, 5529, 6171, 6882, 7638, 8470, 9352, 10318, 11340, 12453, 13629, 14904, 16248, 17700, 19228, 20872, 22600, 24453, 26397, 28476

Offset: 0

Andrew Howroyd, Oct 15 2024

Also, the number of n X 3 {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.

			The a(2) = 4 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]
The a(3) = 6 matrices are:
  [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]  [2 1 0]  [2 0 1]
  [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]  [1 0 2]  [1 2 0]
  [1 1 1]  [1 1 1]  [1 1 1]  [1 1 1]  [0 2 1]  [0 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Column k=3 of A377063.

Cf. A172634, A377065, A377068.

Vec((1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)) + O(x^51))

G.f.: (2/(1 - x^3) - 1)/((1 - x)*(1 - x^2)^3).

G.f.: (1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)).

cp's OEIS Frontend

A208446 Duplicate of A172634.

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A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

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A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^k.

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Mathematica

A328725 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

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A262704 Triangle: Newton expansion of C(n,m)^3, read by rows.

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A377065 Number of 3 X n 0..2 matrices with row sums n and column sums 3 up to permutations of rows.

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A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.

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PARI

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A328747 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Sum_{i=0..n} (-1)^(n-i)binomial(n,i)Sum_{j=0..i} binomial(i,j)^k.

A328725 Constant term in the expansion of (1 + x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.