A143685 Pascal-(1,9,1) array.
1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 141, 31, 1, 1, 41, 361, 361, 41, 1, 1, 51, 681, 1991, 681, 51, 1, 1, 61, 1101, 5921, 5921, 1101, 61, 1, 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1, 1, 81, 2241, 24681, 96201, 96201, 24681, 2241, 81, 1, 1, 91, 2961, 41511, 239241, 460251, 239241, 41511, 2961, 91, 1
Offset: 0
Examples
Square array begins as: 1, 1, 1, 1, 1, 1, 1, ... A000012; 1, 11, 21, 31, 41, 51, 61, ... A017281; 1, 21, 141, 361, 681, 1101, 1621, ... 1, 31, 361, 1991, 5921, 13151, 24681, ... 1, 41, 681, 5921, 29761, 96201, 239241, ... 1, 51, 1101, 13151, 96201, 460251, 1565301, ... 1, 61, 1621, 24681, 239241, 1565301, 7272861, ... Antidiagonal triangle begins as: 1; 1, 1; 1, 11, 1; 1, 21, 21, 1; 1, 31, 141, 31, 1; 1, 41, 361, 361, 41, 1; 1, 51, 681, 1991, 681, 51, 1; 1, 61, 1101, 5921, 5921, 1101, 61, 1; 1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;
Links
- G. C. Greubel, Antidiagonal rows n = 0..50, flattened
Crossrefs
Programs
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Magma
A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >; [A143685(n,k,9): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021
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Mathematica
Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *) -
Sage
flatten([[hypergeometric([-k, k-n], [1], 10).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021
Formula
Square array: T(n, k) = T(n, k-1) + 9*T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(0, k) = 1.
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.
Riordan array (1/(1-x), x*(1+9*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - Jean-François Alcover, May 24 2013
Sum_{k=0..n} T(n, k) = A002534(n+1). - G. C. Greubel, May 29 2021