cp's OEIS Frontend

A247489 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.

%I A247489 #17 Apr 26 2016 12:45:11
%S A247489 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 A247489 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 A247489 13,55,1024,0,0,1,1,1,1,2,4,7,19,89,2048
%N A247489 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 A247489 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 A247489 First few rows of the square array:
%e A247489 [k\n]                                             if conjecture true
%e A247489 [1], 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...     A000079  n>=0
%e A247489 [2], 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...     'A000045' n>=1
%e A247489 [3], 0, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, ...    A000930  n>=4
%e A247489 [4], 0, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, ...     A003269  n>=9
%e A247489 [5], 0, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 11, 15, ...   A003520  n>=16
%e A247489 [6], 0, 1, 1, 1, 1, 1, 2, 3, 3, 4, 6, 7, 10, ...    A005708  n>=25
%e A247489 [7], 0, 0, 1, 1, 1, 1, 1, 2, 3, 3, 4, 5, 7, 8, ...  A005709  n>=36
%e A247489 [8], 0, 0, 1, 1, 1, 1, 2, 1, 2, 3, 3, 4, 5, 6, ...  A005710  n>=49
%e A247489 'A000045' means that the Fibonacci numbers as referenced here start 1, 1, 2, 3, ... for n>=0.
%p A247489 A247489 := proc(k, n)
%p A247489 seq((j-n)/k, j=0..k-1); seq((j-n)/(k-1), j=0..k-2);
%p A247489 hypergeom([%%], [%], -k^k/(k-1)^(k-1));
%p A247489 round(evalf(%,100)) end: # Adjust precision if necessary!
%p A247489 for k from 1 to 9 do print(seq(A247489(k, n), n=0..16)) od;
%o A247489 (Sage)
%o A247489 def A247489(k, n):
%o A247489     P = [(j-n)/k for j in range(k)]
%o A247489     Q = [(j-n)/(k-1) for j in range(k-1)]
%o A247489     H = hypergeometric(P, Q, -k^k/(k-1)^(k-1))
%o A247489     return round(H.n(100)) # Adjust precision if necessary!
%Y A247489 Cf. A000079, A000045, A000930, A003269, A003520, A005708, A005709, A005710.
%K A247489 nonn,tabl
%O A247489 0,3
%A A247489 _Peter Luschny_, Sep 19 2014