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A104698 Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j+1,k+1). 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The n-th column of the triangle = binomial transform of n-th row of A081277, followed by zeros. Example: column 3, (1, 6, 19, 44,...) = b.t. 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; 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

The first few rows are:

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

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: A133807 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

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)