%I #15 May 01 2023 09:28:41
%S 1,1,1,1,3,1,1,17,7,1,1,111,129,15,1,1,769,2815,769,31,1,1,5503,65537,
%T 47103,4097,63,1,1,40193,1579007,3080193,647167,20481,127,1,1,297727,
%U 38862849,208470015,109051905,7929855,98305,255,1,1,2228225,970522623,14413725697,19012780031,3271557121,90177535,458753,511,1
%N Square array read by antidiagonals: T(n,k) = Sum_{j = 0..n*k} binomial(n+j-1,j)*2^j; n,k >= 0.
%C We conjecture that each column sequence satisfies the following supercongruences:
%C Column k: T(n*p^j, k) == T(n*p^(j-1),k) ( mod p^(3*j) ) for prime p >= 5 and positive integers n and j. Some examples are given below.
%F T(n,k) = Sum_{j = 0..n*k} binomial(n+j-1,j)*2^j.
%F Conjectural o.g.f. for column k: 2^(k+1)*x*f'(k,(2^k)*x)/(2*f(k,(2^k)*x) - 1) + 1/(1 + x), where f(k,x) = Sum_{n >= 0} 1/((k+1)*n+1)*C((k+1)*n+1,n)* x^n.
%e Square array begins
%e |k=0 k=1 k=2 k=3 k=4
%e - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e n=0 | 1 1 1 1 1
%e n=1 | 1 3 7 15 31
%e n=2 | 1 17 129 769 4097
%e n=3 | 1 111 2815 47103 647167
%e n=4 | 1 769 65537 3080193 109051905
%e n=5 | 1 5503 1579007 208470015 19012780031
%e n=6 | 1 40193 38862849 14413725697 3385776406529
%e n=7 | 1 297727 970522623 1011196362751 611732191969279
%e ...
%e Examples of congruences for column k = 1:
%e T(5,1) - T(1,1) = 5503 - 3 = (2^2)*(5^3)*11 == 0 ( mod 5^3 ).
%e T(7,1) - T(1,1) = 297727 - 3 = (2^2)*(7^4)*31 == 0 ( mod 7^3 ).
%e T(2*11,1) - T(2,1) = 5913649000782757889 - 17 = (2^4)*(3^2)*(11^3)*107*288357478039 == 0 ( mod 11^3 ).
%e T(5^2,1) - T(5,1) = 2840491845703386005503 - 5503 = (2^7)*(3^3)*(5^6)*7*19*1123*352183001 == 0 ( mod 5^6 ).
%p T := (n, k) -> add(binomial(n+j-1, j)*2^j, j = 0..n*k):
%p T_col := k -> seq(T(n, k), n = 0..7):
%p seq(print(T_col(k)), k = 0..10);
%Y A119259 (column 1), A333561 (column 2), A333562 (column 3). Cf. A333563.
%K nonn,easy,tabf
%O 0,5
%A _Peter Bala_, Mar 26 2020
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