A370707 -id:A370707 - cp's OEIS frontend

A370704 a(n) = Sum_{k=0..n} k!binomial(n, k)Pochhammer(n, k). Row sums of A370707.

1, 2, 17, 442, 23297, 2029226, 262403857, 47086207442, 11184381577217, 3395509635512242, 1282288601819184401, 589443236677619916362, 324023682525763528809217, 209882061169585594259778842, 158200294157346855067204600337, 137282439597406466709932610293026

Offset: 0

Peter Luschny, Feb 28 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..200

Cf. A370707.

a := n -> local k, j; add((-1)^k * mul((j - n)*(j + n), j = 0..k-1), k = 0..n):
seq(a(n), n = 0..15);

A370704[n_] := Sum[k!*Binomial[n, k]*Pochhammer[n, k], {k, 0, n}];
Array[A370704, 20, 0] (* Paolo Xausa, Mar 07 2024 *)

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

a(n) ~ sqrt(Pi) * 4^n * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Mar 12 2024

A380113 Triangle read by rows: The inverse matrix of the central factorials A370707, row n normalized by (-1)^(n - k)*A370707(n, n).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 10, 15, 6, 1, 35, 56, 28, 8, 1, 126, 210, 120, 45, 10, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1

Offset: 0

Peter Luschny, Jan 12 2025

The inverse matrix of A370707 is a rational matrix and the normalization serves to make it a matrix over the integers. Note that the normalization factor A370707(n, n) = FallingFactorial(n, n) * RisingFactorial(n, n) extends A002674 to n = 0.

			Triangle starts:
  [0] [    1]
  [1] [    1,     1]
  [2] [    3,     4,     1]
  [3] [   10,    15,     6,     1]
  [4] [   35,    56,    28,     8,    1]
  [5] [  126,   210,   120,    45,   10,    1]
  [6] [  462,   792,   495,   220,   66,   12,   1]
  [7] [ 1716,  3003,  2002,  1001,  364,   91,  14,   1]
  [8] [ 6435, 11440,  8008,  4368, 1820,  560, 120,  16,  1]
  [9] [24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1]
.
Row 3 of the matrix inverse of the central factorials is [-1/36, 1/24, -1/60, 1/360]. Normalized with (-1)^(n-k)*360 gives row 3 of T.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Variant: A094527.

Cf. A370707, A002674, A008311, A088218 and A110556 (column 0), A081294 (row sums), A000007 (alternating row sums), A005810 (central terms).

T := (n, k) -> if n = k then 1 elif k = 0 then binomial(2*n, n - k)/2 else binomial(2*n, n - k) fi: seq(seq(T(n, k), k = 0..n), n = 0..9);

A380113[n_, k_] := Binomial[2*n, n - k]/(Boole[k == 0 && n > 0] + 1);
Table[A380113[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

def Trow(n):
    def cf(n, k): return falling_factorial(n, k)*rising_factorial(n, k)
    def w(n): return factorial(n)*rising_factorial(n, n)
    m = matrix(QQ, n + 1, lambda x, y: cf(x, y)).inverse()
    return [(-1)^(n-k)*w(n)*m[n, k] for k in range(n+1)]
for n in range(10): print(Trow(n))

T(n, k) = (-1)^(n - k) * ff(n, n) * rf(n, n) * M^(-1)(ff(n, k) * rf(n, k)) where ff denotes the falling factorial, rf the rising factorial and M^(-1)(t(n, k)) the matrix inverse to the matrix with entries t(n, k).
T(n, k) = binomial(2*n, n - k) for 0 < k < n. T(n, n) = 1; T(n, 0) = (-1)^n*binomial(-n, n).
Sum_{k=0..n} T(n, k)*cos(k*x) = 2^(n-1)*(cos(x)+1)^n. (After Philippe Deléham in A008311).

A370706 Triangle read by rows: T(n, k) = binomial(n, k) * Pochhammer(n, k).

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 1, 9, 36, 60, 1, 16, 120, 480, 840, 1, 25, 300, 2100, 8400, 15120, 1, 36, 630, 6720, 45360, 181440, 332640, 1, 49, 1176, 17640, 176400, 1164240, 4656960, 8648640, 1, 64, 2016, 40320, 554400, 5322240, 34594560, 138378240, 259459200

Offset: 0

Peter Luschny, Feb 28 2024

			Triangle starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,    6;
  [3] 1,  9,   36,    60;
  [4] 1, 16,  120,   480,    840;
  [5] 1, 25,  300,  2100,   8400,   15120;
  [6] 1, 36,  630,  6720,  45360,  181440,  332640;
  [7] 1, 49, 1176, 17640, 176400, 1164240, 4656960, 8648640;

Cf. A370707, A000407 (main diagonal), A278070 (row sums).

T := (n, k) -> binomial(n, k)*pochhammer(n, k):
seq(seq(T(n, k), k = 0..n), n = 0..8);

T[n_, k_] := Binomial[n, k] Pochhammer[n, k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = A370707(n, k) / k!.

T(n, n) = Pochhammer(n, n) for n >= 0 (which is different from A000407(n)).

A370983 Triangle read by rows: T(n, k) = (n + k - 1)! / (k!*(n - k)!) if k > 0 and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 20, 0, 4, 30, 120, 210, 0, 5, 60, 420, 1680, 3024, 0, 6, 105, 1120, 7560, 30240, 55440, 0, 7, 168, 2520, 25200, 166320, 665280, 1235520, 0, 8, 252, 5040, 69300, 665280, 4324320, 17297280, 32432400

Offset: 0

Peter Luschny, Mar 07 2024

A signed version of Catalan's triangle (version A128899) can be generated as the scaled inverse of this triangle. The scaled inverse of T is the inverse I of T post-processed by I(n, k) -> I(n, k)/I(n, n).

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 2,   3;
[3] 0, 3,  12,   20;
[4] 0, 4,  30,  120,   210;
[5] 0, 5,  60,  420,  1680,   3024;
[6] 0, 6, 105, 1120,  7560,  30240,  55440;
[7] 0, 7, 168, 2520, 25200, 166320, 665280, 1235520;

Cf. A006963 (main diagonal), A001813 (subdiagonal), A371028 (row sums).

Cf. A370706, A370707, A128899.

T := (n, k) -> `if`(k = 0, k^n, (n + k - 1)! / (k!*(n - k)!)):
seq(seq(T(n, k), k = 0..n), n = 0..9);
A370983 := (n, k) -> local j; ifelse(n = 0, 1, ifelse(k = 0, 0,
(-1)^k*mul((j - n) * (j + n) / (j + 1), j = 0..k - 1) / n)):

T[n_, k_] := If[n == 0, 1, If[k == 0, 0, (n + k - 1)! / (k! * (n - k)!)]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

from math import prod
def T(n, k):
    if n == 0: return 1
    if k == 0: return 0
    return (-1)**k * prod((j - n) * (j + n) / (j + 1) for j in range(k)) / n
for n in range(7): print([T(n, k) for k in range(n + 1)])

def A370983(n, k):
    if k  > n: return 0
    if n == 0: return 1
    if k == 0: return 0
    return binomial(n, k) * rising_factorial(n, k) // n
for n in range(7): print([A370983(n, k) for k in range(n + 1)])

# Added for the sake of reference only.
# For example ScaledInv(A370983, 7) gives the first seven rows of A128899.
def ScaledInv(T, dim): # We assume T(n, n) != 0 for all n.
    M = matrix(QQ, dim, T).inverse()
    for n in range(dim):
        c = M[n][n]
        M[n] = [M.row(n)[k] / c for k in range(dim)]
    return M

Assume n > 0 and k > 0 for the next four formulas:
T(n, k) = ((-1)^k / n) * Product_{j=0..k-1} ((j - n)*(j + n)/(j + 1)).
T(n, k) = binomial(n, k) * Pochhammer(n, k) / n.
T(n, k) = A370706(n, k) / n.
T(n, k) = A370707(n, k) / (n*k!).

cp's OEIS Frontend

A370704 a(n) = Sum_{k=0..n} k!*binomial(n, k)*Pochhammer(n, k). Row sums of A370707.

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A380113 Triangle read by rows: The inverse matrix of the central factorials A370707, row n normalized by (-1)^(n - k)*A370707(n, n).

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A370706 Triangle read by rows: T(n, k) = binomial(n, k) * Pochhammer(n, k).

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Author

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Examples

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Programs

Maple

Mathematica

Formula

A370983 Triangle read by rows: T(n, k) = (n + k - 1)! / (k!*(n - k)!) if k > 0 and T(n, 0) = 0^n.

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A370704 a(n) = Sum_{k=0..n} k!binomial(n, k)Pochhammer(n, k). Row sums of A370707.