A370419 -id:A370419 - cp's OEIS frontend

A370982 a(n) = Sum_{k=0..n} 2^(n - k)*Pochhammer(k/2, n - k). Row sums of A370419(n - k, k).

1, 1, 2, 6, 27, 173, 1464, 15414, 193901, 2834337, 47182518, 880910414, 18225754839, 413835006621, 10230048547292, 273473280803598, 7860479860630329, 241731205891735649, 7919436892637421066, 275351783014543431222, 10126387847107625874803, 392728180939713131370669

Offset: 0

Peter Luschny, Mar 06 2024

nonn

Cf. A370419, A371079, A000522.

a[n_] :=  Sum[2^(n - k)*Pochhammer[k/2, n - k], {k, 0, n}]; Array[a, 22, 0] (* Hugo Pfoertner, Mar 06 2024 *)

From Vaclav Kotesovec, Mar 12 2024: (Start)
Recurrence: (n-4)*a(n) = (4*n^2 - 24*n + 33)*a(n-1) - (2*n - 5)*(2*n^2 - 12*n + 17)*a(n-2) + (4*n^3 - 40*n^2 + 134*n - 151)*a(n-3) - (n-3)*(2*n - 7)*a(n-4).
a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n) * (1 + sqrt(Pi)/(2*sqrt(n))). (End)

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 28, 10, 3, 1, 0, 280, 80, 18, 4, 1, 0, 3640, 880, 162, 28, 5, 1, 0, 58240, 12320, 1944, 280, 40, 6, 1, 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1, 0, 24344320, 4188800, 524880, 58240, 6160, 648, 70, 8, 1

Offset: 0

Werner Schulte and Peter Luschny, Mar 10 2024

			The array starts:
  [0] 1,    1,     1,     1,     1,      1,      1,      1,      1, ...
  [1] 0,    1,     2,     3,     4,      5,      6,      7,      8, ...
  [2] 0,    4,    10,    18,    28,     40,     54,     70,     88, ...
  [3] 0,   28,    80,   162,   280,    440,    648,    910,   1232, ...
  [4] 0,  280,   880,  1944,  3640,   6160,   9720,  14560,  20944, ...
  [5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
  [0] 1;
  [1] 0,       1;
  [2] 0,       1,      1;
  [3] 0,       4,      2,     1;
  [4] 0,      28,     10,     3,    1;
  [5] 0,     280,     80,    18,    4,   1;
  [6] 0,    3640,    880,   162,   28,   5,  1;
  [7] 0,   58240,  12320,  1944,  280,  40,  6, 1;
  [8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
.
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1    1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2    3    4 ... |
    | 0   4   2        | * |     1 3 6 ... | = | 0   4  10   18   28 ... |
    | 0  28  24   6    |   |       1 4 ... |   | 0  28  80  162  280 ... |
    | 0 280 320 144 24 |   |         1 ... |   | 0 280 880 1944 3640 ... |
    | . . .            |   | . . .         |   | . . .                   |

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Family m^n*Pochhammer(k/m, n): A094587 (m=1), A370419 (m=2), this sequence (m=3), A370915 (m=4).
Rows: A000012, A001477, A028552.
Columns: A000007, A007559, A008544, A032031, A007559 (shifted), A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609.
Cf. A303486 (main diagonal), A371079 (row sums of triangle), A371076, A371080.

A := (n, k) -> 3^n*pochhammer(k/3, n):
A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
# Read by antidiagonals:
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
# Using the generating polynomials of the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint([k], EGFcol(k, 8)), k = 0..6);
# As a matrix product:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371076(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));

Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 14 2024 *)

def A(n, k): return 3**n * rising_factorial(k/3, n)
def A(n, k): return (-3)**n * falling_factorial(-k/3, n)

A(n, k) = Product_{j=0..n-1} (3*j + k).
A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
E.g.f. of column k: (1 - 3*t)^(-k/3).
E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371076, i.e., A(n, k) = Sum_{i=0..k} A371076(n, i) * binomial(k, i).

A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 2, 1, 0, 45, 12, 3, 1, 0, 585, 120, 21, 4, 1, 0, 9945, 1680, 231, 32, 5, 1, 0, 208845, 30240, 3465, 384, 45, 6, 1, 0, 5221125, 665280, 65835, 6144, 585, 60, 7, 1, 0, 151412625, 17297280, 1514205, 122880, 9945, 840, 77, 8, 1

Offset: 0

Peter Luschny, Mar 06 2024

The sequence of square arrays A(m, n, k) starts: A094587 (m = 1), A370419 (m = 2), A371077(m = 3), this array (m = 4).

			The array starts:
[0] 1,    1,     1,     1,      1,      1,      1,      1,      1, ...
[1] 0,    1,     2,     3,      4,      5,      6,      7,      8, ...
[2] 0,    5,    12,    21,     32,     45,     60,     77,     96, ...
[3] 0,   45,   120,   231,    384,    585,    840,   1155,   1536, ...
[4] 0,  585,  1680,  3465,   6144,   9945,  15120,  21945,  30720, ...
[5] 0, 9945, 30240, 65835, 122880, 208845, 332640, 504735, 737280, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,      1;
[2] 0,      1,     1;
[3] 0,      5,     2,    1;
[4] 0,     45,    12,    3,   1;
[5] 0,    585,   120,   21,   4,  1;
[6] 0,   9945,  1680,  231,  32,  5, 1;
[7] 0, 208845, 30240, 3465, 384, 45, 6, 1;

Similar square arrays: A094587, A370419, A371077.
Rows: A000012, A001477, A028347, A370914.
Columns: A000007, A007696, A001813, A008545, A047053, A007696, A000407, A034176, A052570 and A034177, A051617, A051618, A051619, A051620.
Cf. A370913 (row sums of triangle), A371026.

A := (n, k) -> 4^n*pochhammer(k/4, n):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 4*x)^(-k/4);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint(EGFcol(n, 9)), n = 0..5);
# Using the generating polynomials for the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-4)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..8)), n = 0..5);
# Implementing the LU decomposition of A:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371026(n-1, k-1)):
U := Matrix(7, 7, (n, k) -> binomial(n-1, k-1)):
MatrixMatrixMultiply(L, Transpose(U));

A[n_, k_] := 4^n * Pochhammer[k/4, n]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

def A(n, k): return 4**n * rising_factorial(k/4, n)
for n in range(6): print([A(n, k) for k in range(9)])

A(n, k) = 4^n*Product_{j=0..n-1} (j + k/4).
A(n, k) = 4^n*Gamma(k/4 + n) / Gamma(k/4) for k >= 1.
The exponential generating function for column k is (1 - 4*x)^(-k/4). But much more is true: (1 - m*x)^(-k/m) are the exponential generating functions for the columns of the arrays A(m, n, k) = m^n*Pochhammer(k/m, n).
The polynomials P(n, x) = Sum_{k=0..n} Stirling1(n, k)*(-4)^(n-k)*x^k are ordinary generating functions for row n, i.e., A(n, k) = P(n, k).
In A370419 Werner Schulte pointed out how A371025 is related to the LU decomposition of A370419. Here the same procedure can be used and amounts to A = A371026 * transpose(binomial triangle), where '*' denotes matrix multiplication. See the Maple section for an implementation.

A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 6, 4, 3, 1, 0, 12, 16, 6, 4, 1, 0, 60, 32, 30, 8, 5, 1, 0, 120, 192, 60, 48, 10, 6, 1, 0, 840, 384, 420, 96, 70, 12, 7, 1, 0, 1680, 3072, 840, 768, 140, 96, 14, 8, 1, 0, 15120, 6144, 7560, 1536, 1260, 192, 126, 16, 9, 1

Offset: 0

Peter Luschny, Mar 04 2024

			The array starts:
[0] 1,  1,   1,   1,   1,    1,    1,    1,    1,    1, ...
[1] 0,  1,   2,   3,   4,    5,    6,    7,    8,    9, ...
[2] 0,  2,   4,   6,   8,   10,   12,   14,   16,   18, ...
[3] 0,  6,  16,  30,  48,   70,   96,  126,  160,  198, ...
[4] 0, 12,  32,  60,  96,  140,  192,  252,  320,  396, ...
[5] 0, 60, 192, 420, 768, 1260, 1920, 2772, 3840, 5148, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   2,   2,  1;
[4] 0,   6,   4,  3,  1;
[5] 0,  12,  16,  6,  4,  1;
[6] 0,  60,  32, 30,  8,  5, 1;
[7] 0, 120, 192, 60, 48, 10, 6, 1;

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Rows: A000012, A001477, A005843, A054000, A134582.
Columns: A000007, A081125, A355989.
Cf. A370419.

A := (n, k) -> 2^n*pochhammer(k/2, iquo(n+1,2)):
for n from 0 to 5 do seq(A(n, k), k = 0..9) od;
T := (n, k) -> A(n - k, k):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A370890[n_, k_] := 2^n*Pochhammer[k/2, Floor[(n+1)/2]];
Table[A370890[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 06 2024 *)

# Note the use of different kinds of division.
def A(n, k): return 2**n * rising_factorial(k/2, (n+1)//2)
for n in range(0, 9): print([A(n, k) for k in range(0, 9)])

A370912 a(n) = n(n + 2)(n + 4).

Original entry on oeis.org

0, 15, 48, 105, 192, 315, 480, 693, 960, 1287, 1680, 2145, 2688, 3315, 4032, 4845, 5760, 6783, 7920, 9177, 10560, 12075, 13728, 15525, 17472, 19575, 21840, 24273, 26880, 29667, 32640, 35805, 39168, 42735, 46512, 50505, 54720, 59163, 63840, 68757, 73920

Offset: 0

Peter Luschny, Mar 05 2024

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cases of A370419(n, k): A000012 (n=0), A001477 (n=1), A005563 (n=2), this sequence (n=3), A190577(n=4).
Cf. A077415, A370890.
Cf. A028347, A028387, A211441.

a := n -> n*(n + 2)*(n + 4): seq(a(n), n = 0..40);
# Using the generating function:
gf := 3*x*(x^2 - 4*x + 5)/(x - 1)^4: ser := series(gf, x, 42):
seq(coeff(ser, x, n), n = 0..40);

Table[n(n+2)(n+4), {n,0,40}] (* or *) CoefficientList[Series[3*x*(x^2 - 4*x + 5)/(x - 1)^4,{x,0,40}],x] (* James C. McMahon, Mar 05 2024 *)

a(n) = 8*Pochhammer(n/2, 3).
a(n) = [x^n] 3*x*(x^2 - 4*x + 5)/(x - 1)^4.
a(n) = 3 * A077415(n + 2).
From Klaus Purath, Aug 02 2024: (Start)
a(n)^2 = A028347(n+2)^3 + 4*A028347(n+2)^2.
a(n+1) - a(n) = A211441(n+2).
a(n) = 3*Sum_{i = 1..n} A028387(i). (End)
E.g.f.: exp(x)*x*(15 + 9*x + x^2). - Stefano Spezia, Aug 18 2024
From Amiram Eldar, Oct 03 2024: (Start)
Sum_{n>=1} 1/a(n) = 11/96.
Sum_{n>=1} (-1)^(n+1)/a(n) = 5/96. (End)

cp's OEIS Frontend

A370982 a(n) = Sum_{k=0..n} 2^(n - k)*Pochhammer(k/2, n - k). Row sums of A370419(n - k, k).

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A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

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A370915 A(n, k) = 4^n*Pochhammer(k/4, n). Square array read by ascending antidiagonals.

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A370890 A(n, k) = 2^n*Pochhammer(k/2, floor((n+1)/2)). Square array read by ascending antidiagonals.

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A370912 a(n) = n*(n + 2)*(n + 4).

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A370912 a(n) = n(n + 2)(n + 4).