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A104698
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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).
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5
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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; internal format)
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OFFSET
| 1,2
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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
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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
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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;
...
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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
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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
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CROSSREFS
| Diagonal sums are A008937(n+1).
Cf. A048739 (row sums), A008288, A005900 (column 3), A014820 (column 4)
Cf. A081277, A142978 by antidiagonals, A119328.
Sequence in context: A133807 A093375 A103283 * A067066 A125103 A171275
Adjacent sequences: A104695 A104696 A104697 * A104699 A104700 A104701
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2005
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EXTENSIONS
| Offset corrected, R. J. Mathar, Sep 04 2011
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