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Square array read by antidiagonals: A(k, n) = hypergeometric(P, Q, -k^k/(k-1)^(k-1)) rounded to the nearest integer, P = [(j-n)/k, j=0..k-1] and Q = [(j-n)/(k-1), j=0..k-2], k>=1, n>=0.
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%I #17 Apr 26 2016 12:45:11

%S 1,0,2,0,1,4,0,1,2,8,0,1,1,3,16,0,1,1,2,5,32,0,1,1,1,3,8,64,0,0,1,1,2,

%T 4,13,128,0,0,1,1,1,3,6,21,256,0,0,1,1,1,2,4,9,34,512,0,0,1,1,1,1,3,5,

%U 13,55,1024,0,0,1,1,1,1,2,4,7,19,89,2048

%N Square array read by antidiagonals: A(k, n) = hypergeometric(P, Q, -k^k/(k-1)^(k-1)) rounded to the nearest integer, P = [(j-n)/k, j=0..k-1] and Q = [(j-n)/(k-1), j=0..k-2], k>=1, n>=0.

%C Conjecture: hypergeometric(P, Q, -k^k/(k-1)^(k-1)) = sum_{j=0.. floor(n/k)} binomial(n-(k-1)*j, j) for n>=(k-1)^2, P and Q as above. (This means for n>=(k-1)^2 the representation is exact without rounding.)

%e First few rows of the square array:

%e [k\n] if conjecture true

%e [1], 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... A000079 n>=0

%e [2], 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... 'A000045' n>=1

%e [3], 0, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, ... A000930 n>=4

%e [4], 0, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, ... A003269 n>=9

%e [5], 0, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 11, 15, ... A003520 n>=16

%e [6], 0, 1, 1, 1, 1, 1, 2, 3, 3, 4, 6, 7, 10, ... A005708 n>=25

%e [7], 0, 0, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, ... A005709 n>=36

%e [8], 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 5, 6, ... A005710 n>=49

%e 'A000045' means that the Fibonacci numbers as referenced here start 1, 1, 2, 3, ... for n>=0.

%p A247489 := proc(k, n)

%p seq((j-n)/k, j=0..k-1); seq((j-n)/(k-1), j=0..k-2);

%p hypergeom([%%], [%], -k^k/(k-1)^(k-1));

%p round(evalf(%,100)) end: # Adjust precision if necessary!

%p for k from 1 to 9 do print(seq(A247489(k, n), n=0..16)) od;

%o (Sage)

%o def A247489(k, n):

%o P = [(j-n)/k for j in range(k)]

%o Q = [(j-n)/(k-1) for j in range(k-1)]

%o H = hypergeometric(P, Q, -k^k/(k-1)^(k-1))

%o return round(H.n(100)) # Adjust precision if necessary!

%Y Cf. A000079, A000045, A000930, A003269, A003520, A005708, A005709, A005710.

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, Sep 19 2014