A065441 -id:A065441 - cp's OEIS frontend

A065432 Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).

1, 1, -1, 1, -2, 0, 1, -3, 1, 1, 1, -4, 3, 2, 0, 1, -5, 6, 2, -2, -2, 1, -6, 10, 0, -6, -4, 0, 1, -7, 15, -5, -11, -3, 5, 5, 1, -8, 21, -14, -15, 4, 15, 10, 0, 1, -9, 28, -28, -15, 19, 26, 6, -14, -14, 1, -10, 36, -48, -7, 42, 30, -16, -42, -28, 0, 1, -11, 45, -75, 14, 70, 16, -60, -70, -14, 42, 42, 1, -12, 55, -110, 54, 96, -28, -120, -70, 56, 126, 84, 0

Offset: 0

Wouter Meeussen, Nov 16 2001

Sums of odd rows are 0, of even rows are the Catalan numbers (A000108) with alternating signs. Row sums of unsigned version give A065441.

			Triangle starts:
[0] 1;
[1] 1, -1;
[2] 1, -2,  0;
[3] 1, -3,  1,   1;
[4] 1, -4,  3,   2,   0;
[5] 1, -5,  6,   2,  -2, -2;
[6] 1, -6, 10,   0,  -6, -4,  0;
[7] 1, -7, 15,  -5, -11, -3,  5,  5;
[8] 1, -8, 21, -14, -15,  4, 15, 10,   0;
[9] 1, -9, 28, -28, -15, 19, 26,  6, -14, -14.

a[0, 0] := 1; a[n_, k_] := 0/;(k > n||n < 0||k < 0); a[n_, k_] := a[n, k] = a[n, k-1]-2a[n-1, k-1]+a[n-1, k]; Table[a[n, k], {n, 0, 16}, {k, 0, n}]

More terms from Sean A. Irvine, Sep 03 2023

A094184 Triangle read by rows in which each term equals the entry above minus the entry left plus twice the entry left-above.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 1, -1, 1, 4, 3, -2, 0, 1, 5, 6, -2, -2, 2, 1, 6, 10, 0, -6, 4, 0, 1, 7, 15, 5, -11, 3, 5, -5, 1, 8, 21, 14, -15, -4, 15, -10, 0, 1, 9, 28, 28, -15, -19, 26, -6, -14, 14, 1, 10, 36, 48, -7, -42, 30, 16, -42, 28, 0, 1, 11, 45, 75, 14, -70, 16, 60, -70, 14, 42, -42, 1, 12, 55, 110, 54, -96, -28, 120, -70, -56, 126, -84, 0, 1

Offset: 0

Wouter Meeussen, May 06 2004

Row sums are A086990 or A090412. (Superseeker finds that the j-th coefficient of OGF(A090412)(z)*(1-z)^j equals A049122). Same absolute values as A065432. Even rows end in 0, odd rows end in Catalan numbers (A000118) with alternating sign.

			Table starts {1},{1,1},{1,2,0},{1,3,1,-1},{1,4,3,-2,0},{1,5,6,-2,-2,2}

Cf. A086990, A090412, A049122, A009766, A065432, A065441.

T[, 0]:=1;T[0, 0]:=1;T[i, j_]/;j>i:=0;T[i_, j_]:=T[i, j]=T[i-1, j]-T[i, j-1]+2 T[i-1, j-1]

T(i, j)=T(i-1, j)-T(i, j-1)+2*T(i-1, j-1), with T(i, 0)=1 and T(i, j)=0 if j>i.

cp's OEIS Frontend

A065432 Triangle related to Catalan triangle: recurrence related to A033877 (Schroeder numbers).

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