

A093190


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


1



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, 1, 14, 93, 292, 365, 186, 49, 8, 1, 16, 129, 544, 1045, 816, 301, 64, 9, 1, 18, 171, 916, 2505, 3006, 1603, 456, 81, 10, 1, 20, 219, 1432, 5225, 9276, 7315, 2864, 657, 100, 11
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

t(k,n) = number of nlong kary words that simultaneously avoid the patterns 112 and 212.


LINKS

G. C. Greubel, Antidiagonal rows n = 1..50, flattened
A. Burstein and T. Mansour, Words restricted by patterns with at most 2 distinct letters, arXiv:math/0110056 [math.CO], 2001.


FORMULA

t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k1, j1). (square array)
T(n, k) = Sum_{j=0..nk+1} j!*binomial(k,j)*binomial(nk,j1). (number triangle)  G. C. Greubel, Mar 09 2021


EXAMPLE

Square array begins as:
1 1 1 1 1 1 ... 1*A000012;
2 4 6 8 10 12 ... 2*A000027;
3 9 21 39 63 93 ... 3*A002061;
4 16 52 136 292 544 ... 4*A135859;
5 25 105 365 1045 2505 ... ;
Antidiagonal rows begins as:
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;


MATHEMATICA

T[n_, k_]:= Sum[j!*Binomial[k, j]*Binomial[nk, j1], {j, 0, nk+1}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)


PROG

(PARI) t(n, k)=sum(j=0, k, j!*binomial(k, j)*binomial(n1, j1))
(Sage) flatten([[ sum(factorial(j)*binomial(k, j)*binomial(nk, j1) for j in (0..nk+1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 09 2021
(Magma) [(&+[Factorial(j)*Binomial(k, j)*Binomial(nk, j1): j in [0..nk+1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 09 2021


CROSSREFS

Main diagonal is A052852.
Antidiagonal sums are in A084261  1.
Sequence in context: A103406 A142978 A152060 * A132191 A094437 A172431
Adjacent sequences: A093187 A093188 A093189 * A093191 A093192 A093193


KEYWORD

nonn,tabl


AUTHOR

Ralf Stephan, Apr 20 2004


STATUS

approved



