A172171 (1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

1, 1, 10, 1, 11, 19, 1, 12, 30, 28, 1, 13, 42, 58, 37, 1, 14, 55, 100, 95, 46, 1, 15, 69, 155, 195, 141, 55, 1, 16, 84, 224, 350, 336, 196, 64, 1, 17, 100, 308, 574, 686, 532, 260, 73, 1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82

Offset: 1

Author

Mark Dols, Jan 28 2010

Keywords

• nonn
• tabl

Comments

Binomial transform of A017173.

Examples

			Triangle begins:
  1;
  1, 10;
  1, 11,  19;
  1, 12,  30,  28;
  1, 13,  42,  58,   37;
  1, 14,  55, 100,   95,   46;
  1, 15,  69, 155,  195,  141,   55;
  1, 16,  84, 224,  350,  336,  196,   64;
  1, 17, 100, 308,  574,  686,  532,  260,   73;
  1, 18, 117, 408,  882, 1260, 1218,  792,  333,   82;
  1, 19, 135, 525, 1290, 2142, 2478, 2010, 1125,  415,  91;
  1, 20, 154, 660, 1815, 3432, 4620, 4488, 3135, 1540, 506, 100;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A007318, A017173, A050489 (central terms), A093644, A139634 (row sums).

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[n==2 && k==2, 10, T[n-1, k] + 2*T[n-1, k-1] - T[n-2, k-1] - T[n-2, k-2]]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 24 2022 *)

SageMath

@CachedFunction
def T(n,k):
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (n==2 and k==2): return 10
    else: return T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 24 2022

Formula

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(n,1) = 1, T(2,2) = 10, T(n,k) = 0 if k < 1 or if k > n.

Sum_{k=0..n} T(n, k) = A139634(n).

T(2*n-1, n) = A050489(n).

Extensions

More terms from Philippe Deléham, Dec 25 2013