A143685 - OEIS

%I #10 Sep 08 2022 08:45:37

%S 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,

%T 681,51,1,1,61,1101,5921,5921,1101,61,1,1,71,1621,13151,29761,13151,

%U 1621,71,1,1,81,2241,24681,96201,96201,24681,2241,81,1,1,91,2961,41511,239241,460251,239241,41511,2961,91,1

%N Pascal-(1,9,1) array.

%H G. C. Greubel, <a href="/A143685/b143685.txt">Antidiagonal rows n = 0..50, flattened</a>

%F 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.

%F Number triangle: T(n,k) = Sum_{j=0..n-k} binomial(n-k,j)*binomial(k,j)*10^j.

%F Riordan array (1/(1-x), x*(1+9*x)/(1-x)).

%F T(n, k) = Hypergeometric2F1([-k, k-n], [1], 10). - _Jean-François Alcover_, May 24 2013

%F Sum_{k=0..n} T(n, k) = A002534(n+1). - _G. C. Greubel_, May 29 2021

%e Square array begins as:

%e   1,  1,    1,     1,      1,       1,        1, ... A000012;

%e   1, 11,   21,    31,     41,      51,       61, ... A017281;

%e   1, 21,  141,   361,    681,    1101,     1621, ...

%e   1, 31,  361,  1991,   5921,   13151,    24681, ...

%e   1, 41,  681,  5921,  29761,   96201,   239241, ...

%e   1, 51, 1101, 13151,  96201,  460251,  1565301, ...

%e   1, 61, 1621, 24681, 239241, 1565301,  7272861, ...

%e Antidiagonal triangle begins as:

%e   1;

%e   1,  1;

%e   1, 11,    1;

%e   1, 21,   21,     1;

%e   1, 31,  141,    31,     1;

%e   1, 41,  361,   361,    41,     1;

%e   1, 51,  681,  1991,   681,    51,    1;

%e   1, 61, 1101,  5921,  5921,  1101,   61,  1;

%e   1, 71, 1621, 13151, 29761, 13151, 1621, 71, 1;

%t Table[Hypergeometric2F1[-k, k-n, 1, 10], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, May 24 2013 *)

%o (Magma)

%o A143685:= func< n,k,q | (&+[Binomial(k, j)*Binomial(n-j, k)*q^j: j in [0..n-k]]) >;

%o [A143685(n,k,9): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 29 2021

%o (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

%Y Cf. A002534, A143680, A143682.

%Y Pascal (1,m,1) array: A123562 (m = -3), A098593 (m = -2), A000012 (m = -1), A007318 (m = 0), A008288 (m = 1), A081577 (m = 2), A081578 (m = 3), A081579 (m = 4), A081580 (m = 5), A081581 (m = 6), A081582 (m = 7), A143683 (m = 8), this sequence (m = 9).

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Aug 28 2008