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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 n-long k-ary 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.

FORMULA

t(n, k) = Sum{j=0..n} j!*C(n, j)*C(k-1, j-1). (square array)

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

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[n-k, j-1], {j, 0, n-k+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(n-1, j-1))

(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

(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

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

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Last modified October 28 07:43 EDT 2021. Contains 348321 sequences. (Running on oeis4.)