cp's OEIS Frontend

A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.

%I A093190 #13 Dec 29 2021 09:53:55
%S A093190 1,1,2,1,4,3,1,6,9,4,1,8,21,16,5,1,10,39,52,25,6,1,12,63,136,105,36,7,
%T A093190 1,14,93,292,365,186,49,8,1,16,129,544,1045,816,301,64,9,1,18,171,916,
%U A093190 2505,3006,1603,456,81,10,1,20,219,1432,5225,9276,7315,2864,657,100,11
%N A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.
%C A093190 t(k,n) = number of n-long k-ary words that simultaneously avoid the patterns 112 and 212.
%H A093190 G. C. Greubel, <a href="/A093190/b093190.txt">Antidiagonal rows n = 1..50, flattened</a>
%H A093190 A. Burstein and T. Mansour, <a href="https://arxiv.org/abs/math/0110056">Words restricted by patterns with at most 2 distinct letters</a>, arXiv:math/0110056 [math.CO], 2001.
%F A093190 t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k-1, j-1). (square array)
%F A093190 T(n, k) = Sum_{j=0..n-k+1} j!*binomial(k,j)*binomial(n-k,j-1). (number triangle) - _G. C. Greubel_, Mar 09 2021
%e A093190 Square array begins as:
%e A093190   1  1   1   1    1    1 ... 1*A000012;
%e A093190   2  4   6   8   10   12 ... 2*A000027;
%e A093190   3  9  21  39   63   93 ... 3*A002061;
%e A093190   4 16  52 136  292  544 ... 4*A135859;
%e A093190   5 25 105 365 1045 2505 ... ;
%e A093190 Antidiagonal rows begins as:
%e A093190   1;
%e A093190   1,  2;
%e A093190   1,  4,  3;
%e A093190   1,  6,  9,   4;
%e A093190   1,  8, 21,  16,   5;
%e A093190   1, 10, 39,  52,  25,  6;
%e A093190   1, 12, 63, 136, 105, 36, 7;
%t A093190 T[n_, k_]:= Sum[j!*Binomial[k, j]*Binomial[n-k, j-1], {j,0,n-k+1}];
%t A093190 Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 09 2021 *)
%o A093190 (PARI) t(n,k)=sum(j=0,k,j!*binomial(k,j)*binomial(n-1,j-1))
%o A093190 (Sage) flatten([[ sum(factorial(j)*binomial(k,j)*binomial(n-k,j-1) for j in (0..n-k+1)) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 09 2021
%o A093190 (Magma) [(&+[Factorial(j)*Binomial(k,j)*Binomial(n-k,j-1): j in [0..n-k+1]]): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 09 2021
%Y A093190 Main diagonal is A052852.
%Y A093190 Antidiagonal sums are in A084261 - 1.
%K A093190 nonn,tabl
%O A093190 1,3
%A A093190 _Ralf Stephan_, Apr 20 2004