A375550 Triangle read by rows: T(m, n, k) = binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1) for m = 4.

1, 6, 1, 25, 7, 1, 88, 32, 8, 1, 280, 120, 40, 9, 1, 832, 400, 160, 49, 10, 1, 2352, 1232, 560, 209, 59, 11, 1, 6400, 3584, 1792, 769, 268, 70, 12, 1, 16896, 9984, 5376, 2561, 1037, 338, 82, 13, 1, 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1

Offset: 0

Author

Peter Luschny, Sep 23 2024

Keywords

• nonn
• tabl

Comments

Triangle T(m,n,k) is a Riordan array of the form ((1-x)^(m-1)*(1-2x)^(-m-1), x/(1-x)), for m = 3. - Igor Victorovich Statsenko, Feb 08 2025

Examples

			Triangle starts:
  [0]     1;
  [1]     6,     1;
  [2]    25,     7,     1;
  [3]    88,    32,     8,    1;
  [4]   280,   120,    40,    9,    1;
  [5]   832,   400,   160,   49,   10,    1;
  [6]  2352,  1232,   560,  209,   59,   11,   1;
  [7]  6400,  3584,  1792,  769,  268,   70,  12,  1;
  [8] 16896,  9984,  5376, 2561, 1037,  338,  82, 13,  1;
  [9] 43520, 26880, 15360, 7937, 3598, 1375, 420, 95, 14, 1;
  ...
Seen as an array of the columns:
  [0] 1,  6, 25,  88,  280,  832,  2352,  6400,  16896, ...
  [1] 1,  7, 32, 120,  400, 1232,  3584,  9984,  26880, ...
  [2] 1,  8, 40, 160,  560, 1792,  5376, 15360,  42240, ...
  [3] 1,  9, 49, 209,  769, 2561,  7937, 23297,  65537, ...
  [4] 1, 10, 59, 268, 1037, 3598, 11535, 34832, 100369, ...
  [5] 1, 11, 70, 338, 1375, 4973, 16508, 51340, 151709, ...
  [6] 1, 12, 82, 420, 1795, 6768, 23276, 74616, 226325, ...
		

Crossrefs

Column k: A055585 (k=0), A001794 (k=1), A001789 (k=2), A027608 (k=3), A055586 (k=4).

Cf. A145018 (diagonal n-2), A375549 (row sums), A049612 (alternating row sums), A122433.

Programs

Maple

T := (m, n, k) -> binomial(n + 1, n - k)*hypergeom([m, k - n], [k + 2], -1);
for n from 0 to 9 do seq(simplify(T(4, n, k)), k = 0..n) od;
# As a binomial sum:
T := (m, n, k) -> add(binomial(m + j, m)*binomial(n + 1, n - (j + k)), j = 0..n-k):
for n from 0 to 9 do [n], seq(T(3, n, k), k = 0..n) od;
# Alternative, generating the array of the columns:
cgf := k -> (1 - x)^(2 - k) / (1 - 2*x)^4:
ser := (k, len) -> series(cgf(k), x, len + 2):
Tcol := (k, len) -> seq(coeff(ser(k, len), x, j), j = 0..len):
seq(lprint([k], Tcol(k, 8)), k = 0..6);

Formula

T(m, n, k) = Sum_{j=0..n-k} binomial(m + j, m)*binomial(n + 1, n - (j + k)) for m = 3.

G.f. of column k: (1 - x)^(2 - k) / (1 - 2*x)^4.