A393138 Triangle read by rows. T(n, k) = Sum_{i=0..k} C(k, i) * C(n - i, floor((n - i) / 2)).

1, 1, 2, 2, 3, 5, 3, 5, 8, 13, 6, 9, 14, 22, 35, 10, 16, 25, 39, 61, 96, 20, 30, 46, 71, 110, 171, 267, 35, 55, 85, 131, 202, 312, 483, 750, 70, 105, 160, 245, 376, 578, 890, 1373, 2123, 126, 196, 301, 461, 706, 1082, 1660, 2550, 3923, 6046

Offset

0,3

Links

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

Formula

T(n, k) = T(n, k-1) + T(n-1, k-1) for k > 0, binomial(n, floor(n / 2)) if k = 0.

Example

Triangle begins:
  [0]   1;
  [1]   1,   2;
  [2]   2,   3,   5;
  [3]   3,   5,   8,  13;
  [4]   6,   9,  14,  22,  35;
  [5]  10,  16,  25,  39,  61,   96;
  [6]  20,  30,  46,  71, 110,  171,  267;
  [7]  35,  55,  85, 131, 202,  312,  483,  750;
  [8]  70, 105, 160, 245, 376,  578,  890, 1373, 2123;
  [9] 126, 196, 301, 461, 706, 1082, 1660, 2550, 3923, 6046;

Maple

T := proc(n, k) option remember; ifelse(k = 0, binomial(n, iquo(n, 2)), T(n, k-1) + T(n-1, k-1)) end: for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

Mathematica

FoldList[Accumulate[Join[{Binomial[#2, Quotient[#2, 2]]}, #]] &, {1}, Range[15]] (* Paolo Xausa, Mar 09 2026 *)

Prog

(Python)
from math import comb as binom
def A393138(n: int, k: int) -> int:
    return sum(binom(k, i) * binom(n-i, (n-i) >> 1) for i in range(k + 1))
for n in range(10): print([A393138(n, k) for k in range(n + 1)])

Crossrefs

T(n, 0) = A001405(n), T(n, n) = A005773(n+1), A393139 (row sums).

Sequence in context: A202874 A355197 A199512 * A303969 A304931 A304669

Adjacent sequences: A393135 A393136 A393137 * A393139 A393140 A393141

Keyword

nonn,tabl

Author

Peter Luschny, Mar 05 2026

Status

approved