A143683 Pascal-(1,8,1) array.
1, 1, 1, 1, 10, 1, 1, 19, 19, 1, 1, 28, 118, 28, 1, 1, 37, 298, 298, 37, 1, 1, 46, 559, 1540, 559, 46, 1, 1, 55, 901, 4483, 4483, 901, 55, 1, 1, 64, 1324, 9856, 21286, 9856, 1324, 64, 1, 1, 73, 1828, 18388, 67006, 67006, 18388, 1828, 73, 1, 1, 82, 2413, 30808, 164242, 304300, 164242, 30808, 2413, 82, 1
Offset: 0
Examples
Square array begins as: 1, 1, 1, 1, 1, 1, 1, ... A000012; 1, 10, 19, 28, 37, 46, 55, ... A017173; 1, 19, 118, 298, 559, 901, 1324, ... 1, 28, 298, 1540, 4483, 9856, 18388, ... 1, 37, 559, 4483, 21286, 67006, 164242, ... 1, 46, 901, 9856, 67006, 304300, 1004590, ... 1, 55, 1324, 18388, 164242, 1004590, 4443580, ... Antidiagonal triangle begins as: 1; 1, 1; 1, 10, 1; 1, 19, 19, 1; 1, 28, 118, 28, 1; 1, 37, 298, 298, 37, 1; 1, 46, 559, 1540, 559, 46, 1; 1, 55, 901, 4483, 4483, 901, 55, 1;
Links
- Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Crossrefs
Programs
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Haskell
a143683 n k = a143683_tabl !! n !! k a143683_row n = a143683_tabl !! n a143683_tabl = map fst $ iterate (\(us, vs) -> (vs, zipWith (+) (map (* 8) ([0] ++ us ++ [0])) $ zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1]) -- Reinhard Zumkeller, Mar 16 2014 -
Magma
A143683:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >; [A143683(n,k,8): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021
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Mathematica
Table[Hypergeometric2F1[-k, k-n, 1, 9], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, May 24 2013 *) -
Sage
flatten([[hypergeometric([-k, k-n], [1], 9).simplify() for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021
Formula
Square array: T(n, 0) = T(0, k) = 1, T(n, k) = T(n, k-1) + 8*T(n-1, k-1) + T(n-1, k).
Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*9^j.
Rows are the expansions of (1+8*x)^k/(1-x)^(k+1).
Riordan array (1/(1-x), x*(1+8*x)/(1-x)).
T(n, k) = Hypergeometric2F1([-k, k-n], [1], 9). - Jean-François Alcover, May 24 2013
E.g.f. for the n-th subdiagonal, n = 0,1,2,..., equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(9*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 18*x + 81*x^2/2) = 1 + 19*x + 118*x^2/2! + 298*x^3/3! + 559*x^4/4! + 901*x^5/5! + .... - Peter Bala, Mar 05 2017
Sum_{k=0..n} T(n,k) = A003683(n+1). - G. C. Greubel, May 27 2021