A357585 -id:A357585 - cp's OEIS frontend

A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

1, 0, 1, -1, 0, 1, -1, -2, 0, 1, 0, -2, -3, 0, 1, 1, 1, -3, -4, 0, 1, 1, 4, 3, -4, -5, 0, 1, 0, 3, 9, 6, -5, -6, 0, 1, -1, -2, 5, 16, 10, -6, -7, 0, 1, -1, -6, -9, 6, 25, 15, -7, -8, 0, 1, 0, -4, -18, -24, 5, 36, 21, -8, -9, 0, 1, 1, 3, -7, -39, -50, 1, 49, 28, -9, -10, 0, 1, 1, 8

Offset: 0

Ralf Stephan, Mar 17 2005

Riordan array ((1-x)/(1-x+x^2),x(1-x)/(1-x+x^2)). - Paul Barry, Jun 21 2008

			1
0,1
-1,0,1
-1,-2,0,1
0,-2,-3,0,1
1,1,-3,-4,0,1
1,4,3,-4,-5,0,1
0,3,9,6,-5,-6,0,1
-1,-2,5,16,10,-6,-7,0,1
-1,-6,-9,6,25,15,-7,-8,0,1

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Mathematics and Computer Science, Part of the series Trends in Mathematics pp 127-139, 2000. [alternative link]
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Colloquium on Mathematics and Computer Science, Versailles, September 2000.

Row sums are A009116 with different signs.
Row sums are A146559(n).
Cf. A005043, A091867, A030528, A000108.

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> A005043(n-1)); # Peter Luschny, Oct 09 2022

T(n,m):=sum(binomial(m,j)*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1),k,0,n)*(-1)^(m-j),j,0,m); /* Vladimir Kruchinin, Apr 08 2011 */

T(n,m) = sum(j=0..m, binomial(m,j)*sum(k=0..n, binomial(k,n-k)*(-1)^(n-k)*binomial(k+j-1,j-1))*(-1)^(m-j)). - Vladimir Kruchinin, Apr 08 2011
T(n,m) = sum(k=ceiling((n-m-1)/2)..n-m, binomial(k+m,m)*binomial(k+1,n-k-m)*(-1)^(n-k-m)). - Vladimir Kruchinin, Dec 17 2011
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 20 2013
T(n+5,n) = (n+1)^2. - Philippe Deléham, Feb 20 2013
From Tom Copeland, Nov 04 2014: (Start)
O.g.f.: G(x,t) = Pinv[Cinv(x),t+1] = Cinv(x) / [1 - (t+1)Cinv(x)] = x*(1-x) / [1-(t+1)x(1-x)] = x + t * x^2 + (-1 + t^2) * x^3 + ..., where Cinv(x)= x * (1-x) is the inverse of C(x) = [1-sqrt(1-4*x)]/2, an o.g.f. for the Catalan numbers A000108 and Pinv(x,t) = -P(-x,t) = x/(1-t*x) is the inverse of P(x,t) = x/(1+x*t).
Ginv(x,t)= C[P[x,t+1]]= C[x/(1+(t+1)x)] = {1-sqrt[1-4*x/(1+(t+1)x)]}/2.
The inverse in x of G(x,t) with t replaced by -t is the o.g.f. of A091867, and G(x,t-1) is a signed version of the (mirrored) Fibonacci polynomials A030528. (End)

A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

Original entry on oeis.org

1, -3, 1, 6, -6, 1, -10, 21, -9, 1, 15, -56, 45, -12, 1, -21, 126, -165, 78, -15, 1, 28, -252, 495, -364, 120, -18, 1, -36, 462, -1287, 1365, -680, 171, -21, 1, 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1, -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1

Offset: 0

Paul Barry, Feb 04 2007

The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

			Triangle begins
    1;
   -3,     1;
    6,    -6,     1;
  -10,    21,    -9,      1;
   15,   -56,    45,    -12,      1;
  -21,   126,  -165,     78,    -15,      1;
   28,  -252,   495,   -364,    120,    -18,     1;
  -36,   462, -1287,   1365,   -680,    171,   -21,     1;
   45,  -792,  3003,  -4368,   3060,  -1140,   231,   -24,   1;
  -55,  1287, -6435,  12376, -11628,   5985, -1771,   300, -27,   1;
   66, -2002, 12870, -31824,  38760, -26334, 10626, -2600, 378, -30, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Inverse is A127898.
Alternating sign version of A127893.
Cf. A001764, A095263, A127896.

[(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

for(n=0, 10, for(k=0,n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).
Sum_{k=0..n} T(n, k) = A127896(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).

Terms a(50) onward added by G. C. Greubel, Apr 29 2018

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A104597 Triangle T read by rows: inverse of Motzkin triangle A097609.

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A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

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