This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363418 #25 Jan 05 2024 14:28:00
%S A363418 2,2,8,2,16,38,2,24,146,192,2,32,326,1408,1002,2,40,578,4672,14002,
%T A363418 5336,2,48,902,11008,69002,142000,28814,2,56,1298,21440,216002,
%U A363418 1038984,1459810,157184,2,64,1766,36992,525002,4320608,15856206,15158272,864146
%N A363418 Square array read by ascending antidiagonals: T(n,k) = [x^(n*k)] ((1 + x)/(1 - x))^k for n, k >= 1.
%C A363418 The n-th row sequence {T(n, k) : k >= 1} satisfies the Gauss congruences, that is, T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^r ) for all primes p and positive integers m and r.
%C A363418 We conjecture that each row sequence satisfies the stronger supercongruences T(n, m*p^r) == T(n, m*p^(r-1)) ( mod p^(3*r) ) for all primes p >= 5 and positive integers m and r.
%D A363418 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A363418 Andrew Howroyd, <a href="/A363418/b363418.txt">Table of n, a(n) for n = 1..1275</a> (first 50 antidiagonals)
%F A363418 T(n,k) = Sum_{j = 0..k} binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j).
%F A363418 T(n,k) = 1/n * [x^k] ((1 + x)/(1 - x))*(n*k).
%F A363418 T(n,k) = (1/n)*Sum_{j = 0..k} binomial(n*k, j)*binomial((n + 1)*k - j - 1, k - j).
%F A363418 T(2*n,k) = [x^(n*k)] Chebyshev_T(k,(1 + x)/(1 - x)), where Chebyshev_T(n,x) denotes the n-th Chebyshev polynomial of the first kind. See A053120.
%F A363418 T(n,k) = Sum_{j = 1..k} (2^j)*binomial(k, j)*binomial(n*k - 1, n*k - j).
%F A363418 T(n,k) = (2*k) * hypergeom([1 - k, 1 - n*k], [2], 2).
%F A363418 Define E(n,x) = exp( Sum_{j >= 1} T(n,j)*x^j/j ). Then T(n+1,k) = [x^k] E(n,x)^k.
%F A363418 E(n,x) = (1/x) * the series reversion of x/E(n-1,x) for n >= 2.
%F A363418 E(n,x)^n = (1/x) * the series reversion of x*((1 - x)/(1 + x))^n.
%F A363418 E(m,x) appears to be the g.f. of the (m + 1)-Schroeder numbers. See A027307 (m = 2) and the cross references there.
%F A363418 The o.g.f. for row n is the diagonal of the bivariate rational function (1/n) * t*f(x)^n/(1 - t*f(x)^n), where f(x) = (1 + x)/(1 - x), and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
%e A363418 Square array begins
%e A363418 n\k | 1 2 3 4 5 6 7
%e A363418 - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A363418 1 | 2 8 38 192 1002 5336 28814 ... (A002003)
%e A363418 2 | 2 16 146 1408 14002 142000 1459810 ... (A103885)
%e A363418 3 | 2 24 326 4672 69002 1038984 15856206 ... (A333715)
%e A363418 4 | 2 32 578 11008 216002 4320608 87588482 ...
%e A363418 5 | 2 40 902 21440 525002 13104184 331482062 ...
%e A363418 6 | 2 48 1298 36992 1086002 32497680 985524066 ...
%e A363418 7 | 2 56 1766 58688 2009002 70097384 2478629134 ...
%e A363418 8 | 2 64 2306 87552 3424002 136485568 5513464322 ...
%p A363418 # display as a square array
%p A363418 T := (n,k) -> add( binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j) , j = 0..k): for n from 1 to 10 do seq(T(n, k), k = 1..10) end do;
%p A363418 #alternative program
%p A363418 seq(print(seq(simplify(2*k*hypergeom([1 - k, 1 - n*k], [2], 2)), k = 1..10)), n = 1..10);
%p A363418 # display as a sequence
%p A363418 seq(seq(T(n+1-i, i), i = 1..n), n = 1..10);
%o A363418 (PARI) T(n,k) = sum(j=0, k, binomial(k, j)*binomial((n + 1)*k - j - 1, n*k - j)) \\ _Andrew Howroyd_, Jan 05 2024
%Y A363418 A002003 (row 1), A103885 (row 2), A333715 (row 3). Cf. A035607, A362724 - A362733, A363419.
%K A363418 nonn,tabl,easy
%O A363418 1,1
%A A363418 _Peter Bala_, Jun 12 2023