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Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).
8

%I #31 Mar 10 2019 16:27:22

%S 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,

%T 49,146,225,180,73,14,1,9,64,231,456,501,304,99,16,1,10,81,344,833,

%U 1182,985,476,129,18,1,11,100,489,1408,2471,2668,1765,704,163,20,1,12

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

%C 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

%C 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

%H Reinhard Zumkeller, <a href="/A104698/b104698.txt">Rows n = 1..100 of triangle, flattened</a>

%F 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.

%F 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

%F 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

%F 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

%e Triangle begins

%e 1;

%e 2, 1;

%e 3, 4, 1;

%e 4, 9, 6, 1;

%e 5, 16, 19, 8, 1;

%e 6, 25, 44, 33, 10, 1;

%e 7, 36, 85, 96, 51, 12, 1;

%e ...

%p A104698 := proc(n,k) add(binomial(k,j)*binomial(n-j+1,n-k-j),j=0..n-k) ; end proc:

%p seq(seq(A104698(n,k),k=0..n),n=0..15) ; # _R. J. Mathar_, Sep 04 2011

%t u[1, _] = 1; v[1, _] = 1;

%t u[n_, x_] := u[n, x] = x u[n-1, x] + v[n-1, x] + 1;

%t v[n_, x_] := v[n, x] = 2 x u[n-1, x] + v[n-1, x] + 1;

%t Table[CoefficientList[u[n, x], x], {n, 1, 11}] // Flatten (* _Jean-François Alcover_, Mar 10 2019, after _Clark Kimberling_ *)

%o (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

%o (Haskell)

%o a104698 n k = a104698_tabl !! (n-1) !! (k-1)

%o a104698_row n = a104698_tabl !! (n-1)

%o a104698_tabl = [1] : [2,1] : f [1] [2,1] where

%o f us vs = ws : f vs ws where

%o ws = zipWith (+) ([0] ++ us ++ [0]) $

%o zipWith (+) ([1] ++ vs) (vs ++ [0])

%o -- _Reinhard Zumkeller_, Jul 17 2015

%Y Diagonal sums are A008937(n+1).

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

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

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Mar 19 2005

%E Offset corrected by _R. J. Mathar_, Sep 04 2011