A064339 -id:A064339 - cp's OEIS frontend

A064334 Triangle composed of generalized Catalan numbers.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, 1, 1, 1, -2, 5, -2, 1, 1, 1, 6, -25, 13, -3, 1, 1, 1, -18, 141, -98, 25, -4, 1, 1, 1, 57, -849, 826, -251, 41, -5, 1, 1, 1, -186, 5349, -7448, 2817, -514, 61, -6, 1, 1, 1, 622, -34825, 70309, -33843, 7206, -917, 85, -7, 1, 1

Offset: 0

Wolfdieter Lang, Sep 21 2001

The sequence for column m (m >= 1) (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=:Y_{N}(N+1), N >=0, for (unphysical) alpha = -m, beta = 1 (or alpha = 1, beta = -m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1. See also Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.

			Triangle starts:
  1;
  1,  1;
  1,  1,  1;
  1,  0,  1,  1;
  1,  1, -1,  1, 1;
  1, -2,  5, -2, 1, 1; ...

T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.

G. C. Greubel, Rows n =0..100 of triangle, flattened
B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

The unsigned column sequences (without leading zeros) are A000012, A064310-11, A064325-33 for m=0..11, respectively. Row sums (signed) give A064338. Row sums (unsigned) give A064339.

Cf. A064062.

[[k eq 0 select 1 else k eq n select 1 else (&+[(n-k-j)* Binomial(n-k-1+j, j)*(-k)^j/(n-k): j in [0..n-k-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 04 2019

Table[If[k==0, 1, If[k==n, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]*(-k)^j/(n -k), {j, 0, n-k-1}]]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 04 2019 *)

{T(n,k) = if(k==0, 1, if(k==n, 1, sum(j=0, n-k-1, (n-k-j)* binomial(n-k-1+j, j)*(-k)^j/(n-k))))}; \\ G. C. Greubel, May 04 2019

def T(n,k):
    return hypergeometric([1-n, n], [-n], -k) if n>0 else 1
for n in (0..10):
    print([simplify(T(n-k,k)) for k in (0..n)]) # Peter Luschny, Nov 30 2014

G.f. for column m: (x^m)/(1-x*c(-m*x))= (x^m)*((m+1)+m*x*c(-m*x))/((m+1)-x), m>0, with the g.f. c(x) of Catalan numbers A000108.
T(n, m) = Sum_{k=0..n-m-1} (n-m-k)*binomial(n-m-1+k, k)*(-m)^k/(n-m), with T(n,0) = T(n,n)=1.
T(n,m) = (1/(1+m))^(n-m)*(1 + m*Sum_{k=0..n-m-1} C(k)*(-m*(m+1))^k ), n-m >= 1, T(n, n) = T(n,0) =1, T(n, m)=0 if nA000108(k) (Catalan).
T(n, k) = hypergeometric([1-n+k, n-k], [-n+k], -k) if kPeter Luschny, Nov 30 2014

A064338 Row sums of signed triangle A064334.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, -6, 49, -178, 76, 8633, -124131, 1202662, -8252662, 16009623, 730544913, -17236420029, 256679586178, -2787185606474, 15947981601793, 215110909342463, -9723413157539188, 216257810122284716, -3515999949642686709

Offset: 0

Wolfdieter Lang, Sep 21 2001

sign

Robert Israel, Table of n, a(n) for n = 0..488

Cf. A064339 (row sums of unsigned triangle A064334).

f:= proc(n) local k; 1 + add(simplify(hypergeom([1-n+k,n-k],[-n+k],-k)),k=0..n-1) end proc:
map(f, [$0..50]); # Robert Israel, Jan 03 2019

a[n_] := If[n == 0, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]* (-k)^j/(n-k), {k, 1, n-1}, {j, 0, n-k-1}] + 2];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 06 2023 *)

a(n) = 1 + Sum_{0<=k<=n-1} hypergeom([1-n+k,n-k],[-n+k],-k). - Robert Israel, Jan 03 2019

A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

Original entry on oeis.org

1, 3, 4, 5, 8, 21, 7, 12, 35, 96, 9, 16, 49, 144, 410, 11, 20, 63, 192, 574, 1680, 13, 24, 77, 240, 738, 2240, 6692, 15, 28, 91, 288, 902, 2800, 8604, 26112, 17, 32, 105, 336, 1066, 3360, 10516, 32640, 100296, 19, 36, 119, 384, 1230, 3920, 12428, 39168, 122584, 380480

Offset: 0

Oboifeng Dira, Jul 14 2020

			Triangle begins:
  1;
  3,  4;
  5,  8, 21;
  7, 12, 35,  96;
  9, 16, 49, 144, 410;
  ...
T(3,2) = ((2+sqrt(2))^3-(2-sqrt(2))^3)*(6-2+1)/(4*sqrt(2)) = (28*sqrt(2))*(5)/(4*sqrt(2)) = 35.

Oboifeng Dira, Table of n, a(n) for n = 0..54

Cf. A053199, A294285, A064339.

T := proc (n, k) if k = 0 and 0 <= n then 2*n+1 elif 1 <= k and k <= n then round((((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T(n,k) = if (k==0, 2*n+1, if (k<=n, sum(i=n-k, n, sum(j=0, i-n+k, if ((i==n) && (j==k), 0, T(i,j)), 0))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 08 2020

T(n, k) = if (k==0, 2*n+1, if (k>n, 0, my(w=quadgen(8, 'w)); ((2+w)^(k+1)-(2-w)^(k+1))*(2*n-k+1)/(4*w)));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 10 2020

T(n,0) = 2*n+1 for k=0;

T(n,k) = ((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)) for 1<=k<=n.

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A064334 Triangle composed of generalized Catalan numbers.

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A064338 Row sums of signed triangle A064334.

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A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

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