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

1,3

Comments

Binomial transform of A017173.

Links

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

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).

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A297418 A297352 A370401 * A327723 A377164 A164899

Adjacent sequences: A172168 A172169 A172170 * A172172 A172173 A172174

Keyword

nonn,tabl

Author

Mark Dols, Jan 28 2010

Extensions

More terms from Philippe Deléham, Dec 25 2013

Status

approved