A144431 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n,1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = -1.

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, 2, -2, 1, 1, -3, 2, 2, -3, 1, 1, -4, 7, -8, 7, -4, 1, 1, -5, 9, -5, -5, 9, -5, 1, 1, -6, 16, -26, 30, -26, 16, -6, 1, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 1, -8, 29, -64, 98, -112, 98, -64, 29, -8, 1, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1

Offset: 1

Author

Roger L. Bagula, Oct 04 2008

Keywords

• tabl
• easy
• sign

Comments

Row sums are: {1, 2, 2, 0, 0, 0, 0, 0, 0, 0, ...}.

For m = ...,-1,0,1,2 we get ..., A144431, A007318 (Pascal), A008292, A060187, ..., so this might be called a sub-Pascal triangle.

The triangle starts off like A098593, but is different further on.

Examples

			Triangle begins:
  1;
  1,   1;
  1,   0,   1;
  1,  -1,  -1,   1;
  1,  -2,   2,  -2,   1;
  1,  -3,   2,   2,  -3,   1;
  1,  -4,   7,  -8,   7,  -4,   1;
  1,  -5,   9,  -5,  -5,   9,  -5,   1;
  1,  -6,  16, -26,  30, -26,  16,  -6,   1;
  1,  -7,  20, -28,  14,  14, -28,  20,  -7,   1;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.

Crossrefs

Cf. A007318, A008292, A060187, A098593.

Programs

Maple

T:=proc(n,k,l) option remember;
if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1,k-1,l)+(l*k-l+1)*T(n-1,k,l); fi; end;
for n from 1 to 15 do lprint([seq(T(n,k,-1),k=1..n)]); od; # N. J. A. Sloane, May 08 2013

Mathematica

m=-1;
T[n_, 1]:= 1; T[n_, n_]:= 1;
T[n_, k_]:= (m*n-m*k+1)*T[n-1, k-1] + (m*k - (m - 1))*T[n-1,k];
Table[T[n, k], {n,15}, {k,n}]//Flatten

Sage

def A144431(n,k):
    if (n<3): return 1
    else: return (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3)
flatten([[A144431(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 01 2022

Formula

T(n,k) = (m*n - m*k + 1)*T(n-1, k-1) + (m*k - (m-1))*T(n-1, k) with T(n, 1) = T(n, n) = 1 and m = -1.

From G. C. Greubel, Mar 01 2022: (Start)

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

T(n, k) = (-1)^(k-1)*binomial(n-3, k-1) + (-1)^(n+k)*binomial(n-3, k-3) with T(1, k) = T(2, k) = 1.

Sum_{k=1..n} T(n, k) = [n==1] + 2*[n==2] + 2*[n==3] + (1-(-1)^n)*0^(n-3)*[n>3]. (End)

Extensions

Edited by N. J. A. Sloane, May 08 2013