A128710 Triangle read by rows: T(n,k) = (k+2)*binomial(n,k) (0 <= k <= n).

2, 2, 3, 2, 6, 4, 2, 9, 12, 5, 2, 12, 24, 20, 6, 2, 15, 40, 50, 30, 7, 2, 18, 60, 100, 90, 42, 8, 2, 21, 84, 175, 210, 147, 56, 9, 2, 24, 112, 280, 420, 392, 224, 72, 10, 2, 27, 144, 420, 756, 882, 672, 324, 90, 11, 2, 30, 180, 600, 1260, 1764, 1680, 1080, 450, 110, 12, 2, 33

Offset: 0

Author

David Hoek (david.hok(AT)telia.com), Mar 23 2007

Keywords

• nonn
• tabl

Comments

k*binomial(n-4, k-2) counts the permutations in S_n which have zero occurrences of the pattern 213 and one occurrence of the pattern 132 and k descents.

Sum of row n =(n+4)*2^(n-1) (A045623). - Emeric Deutsch, Apr 02 2007

Essentially the same as A127954: obtained by dropping the first row of A127954. - Peter Bala, Mar 05 2013

Examples

			Triangle starts:
  2;
  2,  3;
  2,  6,  4;
  2,  9, 12,  5;
  2, 12, 24, 20,  6;
		

References

• D. Hök, Parvisa mönster i permutationer [Swedish], (2007).

Crossrefs

Cf. A045623, A127954.

Programs

Maple

T:=(n,k)->(k+2)*binomial(n,k): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form. - Emeric Deutsch, Apr 02 2007

Mathematica

T[n_,k_]:=(k+2)*Binomial[n,k];Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* James C. McMahon, Jan 11 2025 *)

Formula

G.f.: (2 - t*(2+x))/(1 - t*(1+x))^2 = 2 + (2+3*x)*t + (2+6*x+4*x^2)*t^2 + .... - Peter Bala, Mar 05 2013

Row n is the vector of polynomial coefficients of (2 + (n+2)*x)*(1+x)^(n-1). - Peter Bala, Mar 05 2013

Extensions

Edited by Emeric Deutsch, Apr 02 2007