A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).

1, 2, 1, 3, 4, 1, 4, 9, 6, 1, 5, 16, 19, 8, 1, 6, 25, 44, 33, 10, 1, 7, 36, 85, 96, 51, 12, 1, 8, 49, 146, 225, 180, 73, 14, 1, 9, 64, 231, 456, 501, 304, 99, 16, 1, 10, 81, 344, 833, 1182, 985, 476, 129, 18, 1, 11, 100, 489, 1408, 2471, 2668, 1765, 704, 163, 20, 1, 12

Offset

1,2

Comments

The n-th column of the triangle is the binomial transform of the n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44, ...) = binomial transform of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0, ...). A104698 = reversal by rows of A142978. - Gary W. Adamson, Jul 17 2008

This sequence is jointly generated with A210222 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) + 1 and v(n,x) = 2x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section at A210222. - Clark Kimberling, Mar 19 2012

Links

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Formula

The triangle is extracted from the product A * B; A = [1; 1, 1; 1, 1, 1; ...], B = [1; 1, 1; 1, 3, 1; 1, 5, 5, 1; ...] both infinite lower triangular matrices (rest of the terms are zeros). The triangle of matrix B by rows = A008288, Delannoy numbers.

Riordan array (1/(1-x)^2, x(1+x)/(1-x))=(1/(1-x), x)*(1/(1-x), x(1+x)/(1-x)); T(n, k)=sum{j=0..n, sum{i=0..j-k, C(j-k, i)*C(k, i)*2^i}}; T(n, k)=sum{j=0..k, sum{i=n-k-j, (n-k-j-i+1)*C(k, j)*C(k+i-1, i)}}. - Paul Barry, Jul 18 2005

T(n,k) = binomial(n+1,k+1)*2F1(-k,k-n;-n-1;-1) where 2F1(.;.;.) is a Gaussian hypergeometric function. - R. J. Mathar, Sep 04 2011

T(n,1)=n; T(n,n)=1; for 1 < k < n, T(n,k) = T(n-2,k-1) + T(n-1,k-1) + T(n-1,k). - Reinhard Zumkeller, Jul 17 2015

Example

Triangle begins
  1;
  2,  1;
  3,  4,  1;
  4,  9,  6,  1;
  5, 16, 19,  8,  1;
  6, 25, 44, 33, 10,  1;
  7, 36, 85, 96, 51, 12,  1;
  ...

Maple

A104698 := proc(n, k) add(binomial(k, j)*binomial(n-j+1, n-k-j), j=0..n-k) ; end proc:
seq(seq(A104698(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Sep 04 2011

Mathematica

u[1, _] = 1; v[1, _] = 1;
u[n_, x_] := u[n, x] = x u[n-1, x] + v[n-1, x] + 1;
v[n_, x_] := v[n, x] = 2 x u[n-1, x] + v[n-1, x] + 1;
Table[CoefficientList[u[n, x], x], {n, 1, 11}] // Flatten (* Jean-François Alcover, Mar 10 2019, after Clark Kimberling *)

Prog

(PARI) T(n, k)=sum(j=0, n-k, binomial(k, j)*binomial(n-j+1, k+1)) \\ Charles R Greathouse IV, Jan 16 2012
(Haskell)
a104698 n k = a104698_tabl !! (n-1) !! (k-1)
a104698_row n = a104698_tabl !! (n-1)
a104698_tabl = [1] : [2, 1] : f [1] [2, 1] where
   f us vs = ws : f vs ws where
     ws = zipWith (+) ([0] ++ us ++ [0]) $
          zipWith (+) ([1] ++ vs) (vs ++ [0])
-- Reinhard Zumkeller, Jul 17 2015

Crossrefs

Diagonal sums are A008937(n+1).

Cf. A048739 (row sums), A008288, A005900 (column 3), A014820 (column 4)

Cf. A081277, A142978 by antidiagonals, A119328, A110271 (matrix inverse).

Sequence in context: A325001 A093375 A103283 * A067066 A210219 A125103

Adjacent sequences: A104695 A104696 A104697 * A104699 A104700 A104701

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 19 2005

Extensions

Offset corrected by R. J. Mathar, Sep 04 2011

Status

approved