A093190 Array t read by antidiagonals: number of {112,212}-avoiding words.
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
Offset: 1
Examples
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;
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.
Programs
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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
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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 *) -
PARI
t(n,k)=sum(j=0,k,j!*binomial(k,j)*binomial(n-1,j-1))
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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
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
Comments