A090192 -id:A090192 - cp's OEIS frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

1, 1, -10, -1231, 1636130, 23957879562, -3858392581773300, -6835385537899011365535, 133202313157282627679850238250, 28553099061411464607955930776882965774

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 10*x^2 - 1231*x^3 + 1636130*x^4 + 23957879562*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..44
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. this sequence (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=11 of A290789.

m = 10; ContinuedFractionK[If[i == 1, 1, -(-11)^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015108(n)
  A(-11, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-11 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-11*x)) = 1/(1-x/(1+11*x/(1-11^2*x/(1+11^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

A105523 Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, -1, 0, 1, 0, -2, 0, 5, 0, -14, 0, 42, 0, -132, 0, 429, 0, -1430, 0, 4862, 0, -16796, 0, 58786, 0, -208012, 0, 742900, 0, -2674440, 0, 9694845, 0, -35357670, 0, 129644790, 0, -477638700, 0, 1767263190, 0

Offset: 0

Paul Barry, Apr 11 2005

Row sums of A105522. Row sums of inverse of A105438.

First column of number triangle A106180.

			G.f. = 1 - x + x^3 - 2*x^5 + 5*x^7 - 14*x^9 + 42*x^11 - 132*x^13 + 429*x^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 313.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.

Cf. A000108, A097331, A090192, A090181, A105522, A105438.

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 + 2*x - Sqrt(1+4*x^2))/(2*x))); // G. C. Greubel, Sep 16 2018

A105523_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=-a[w-1]+(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A105523_list(40); # Peter Luschny, May 19 2011

a[n_?EvenQ] := 0; a[n_?OddQ] := 4^n*Gamma[n/2] / (Gamma[-n/2]*(n+1)!); a[0] = 1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + 2 x - Sqrt[1 + 4 x^2])/(2 x), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 01 2014 *)
a[ n_] := SeriesCoefficient[ (1 + 2 x - Sqrt[ 1 + 4 x^2]) / (2 x), {x, 0, n}]; (* Michael Somos, Jun 17 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], a[n] = -2 a[n - 1] + Sum[ a[j] a[n - j - 1], {j, 0, n - 1}]]; (* Michael Somos, Jun 17 2015 *)

{a(n) = local(A); if( n<0, 0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = -2 * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

def A105523(n):
    if is_even(n): return 0 if n>0 else 1
    return -(sqrt(pi)*2^(n-1))/(gamma(1-n/2)*gamma((n+3)/2))
[A105523(n) for n in (0..29)] # Peter Luschny, Oct 31 2014

G.f.: (1 + 2*x - sqrt(1+4*x^2))/(2*x).
a(n) = 0^n + sin(Pi*(n-2)/2)(C((n-1)/2)(1-(-1)^n)/2).
G.f.: 1/(1+x/(1-x/(1+x/(1-x/(1+x/(1-x.... (continued fraction). - Paul Barry, Jan 15 2009
a(n) = Sum{k = 0..n} A090181(n,k)*(-1)^k. - Philippe Deléham, Feb 02 2009
a(n) = (1/n)*sum_{i = 0..n-1} (-2)^i*binomial(n, i)*binomial(2*n-i-2, n-1). - Vladimir Kruchinin, Dec 26 2010
With offset 1, a(n) = -2 * a(n-1) + Sum_{k=1..n-1} a(k) * a(n-k), for n>1. - Michael Somos, Jul 25 2011
D-finite with recurrence: (n+3)*a(n+2) = -4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 18 2014
For nonzero terms, a(n) ~ (-1)^((n+1)/2)/sqrt(2*Pi)*2^(n+1)/(n+1)^(3/2). - Fung Lam, Mar 17 2014
a(n) = -(sqrt(Pi)*2^(n-1))/(Gamma(1-n/2)*Gamma((n+3)/2)) for n odd. - Peter Luschny, Oct 31 2014
From Peter Bala, Apr 20 2024: (Start)
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n + k, 2*k)*Catalan(k), where Catalan(k) = A000108(k).
a(n) = (-2)^n * hypergeom([-n, n+1], [2], 1/2).
O.g.f.: A(x) = 1/x * series reversion of x*(1 - x)/(1 - 2*x).  Cf. A152681. (End)

Typo in definition corrected by Robert Israel, Oct 31 2014

A290789 A(n,k) is the n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -7, 0, 1, 1, 1, -3, -23, 47, 2, 1, 1, 1, -4, -55, 586, 873, 0, 1, 1, 1, -5, -109, 3429, 48778, -26433, -5, 1, 1, 1, -6, -191, 13436, 885137, -11759396, -1749159, 0, 1, 1, 1, -7, -307, 40915, 8425506, -904638963, -8596478231, 220526159, 14, 1

Offset: 0

Alois P. Heinz, Aug 10 2017

			Square array A(n,k) begins:
  1,  1,   1,     1,      1,       1, ...
  1,  1,   1,     1,      1,       1, ...
  1,  0,  -1,    -2,     -3,      -4, ...
  1, -1,  -7,   -23,    -55,    -109, ...
  1,  0,  47,   586,   3429,   13436, ...
  1,  2, 873, 48778, 885137, 8425506, ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

Columns k=0-11 give: A000012, A090192, A015097, A015098, A015099, A015100, A015102, A015103, A015105, A015106, A015107, A015108.
Main diagonal gives A290786.
Cf. A290759.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j, k)*A(n-j-1, k)*(-k)^j, j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];A[n_, k_]:=A[n, k]=If[n==0 , 1, Sum[A[j, k] A[n - j - 1, k]* (-k)^j, {j, 0, n - 1}]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] (* Indranil Ghosh, Aug 13 2017 *)

from sympy.core.cache import cacheit
@cacheit
def A(n, k):
    return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*(-k)**j for j in range(n))
for d in range(16): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 13 2017

G.f. of column k: 1/(1-x/(1+k*x/(1-k^2*x/(1+k^3*x/(1-k^4*x/(1+ ... )))))).

A(n,k) = Sum_{j=0..n-1} A(j,k)*A(n-j-1,k)*(-k)^j for n>0, A(0,k) = 1.

A097331 Expansion of 1 + 2x/(1 + sqrt(1 - 4x^2)).

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132, 0, 429, 0, 1430, 0, 4862, 0, 16796, 0, 58786, 0, 208012, 0, 742900, 0, 2674440, 0, 9694845, 0, 35357670, 0, 129644790, 0, 477638700, 0, 1767263190, 0, 6564120420, 0, 24466267020, 0, 91482563640, 0, 343059613650, 0

Offset: 0

Paul Barry, Aug 05 2004

Binomial transform is A097332. Second binomial transform is A014318.
Essentially the same as A126120. - R. J. Mathar, Jun 15 2008
Hankel transform is A087960(n) = (-1)^binomial(n+1,2). - Paul Barry, Aug 10 2009

Michael De Vlieger, Table of n, a(n) for n = 0..3340
Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46.

A097331_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w]:=a[w-1]-(-1)^w*add(a[j]*a[w-j-1],j=1..w-1) od; convert(a,list)end: A097331_list(48); # Peter Luschny, May 19 2011

a[0] = 1; a[n_?OddQ] := CatalanNumber[(n-1)/2]; a[] = 0; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover, Jul 24 2013 *)

def A097331_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append(abs(D[1]))
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A097331_list(49) # Peter Luschny, Jun 03 2012

a(n) = 0^n + Catalan((n-1)/2)(1-(-1)^n)/2.
Unsigned version of A090192, A105523. - Philippe Deléham, Sep 29 2006
From Paul Barry, Aug 10 2009: (Start)
G.f.: 1+xc(x^2), c(x) the g.f. of A000108;
G.f.: 1/(1-x/(1+x/(1+x/(1-x/(1-x/(1+x/(1+x/(1-x/(1-x/(1+... (continued fraction);
G.f.: 1+x/(1-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). (End)
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+2*x) (continued fraction); more generally g.f. C(x/(1+2*x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
Conjecture: (n+1)*a(n) + n*a(n-1) + 4*(-n+2)*a(n-2) + 4*(-n+3)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
Recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=a(1)=1. For nonzero terms, a(n) ~ 2^(n+1)/((n+1)^(3/2)*sqrt(2*Pi)). - Fung Lam, Mar 17 2014

A364759 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 4, 25, 182, 1447, 12175, 106575, 960579, 8854622, 83089537, 791063172, 7622317663, 74191096721, 728389554533, 7204640725610, 71727367291455, 718195853746770, 7227785937663908, 73069500402699226, 741712341691454837, 7556704348506425398

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Cf. A090192, A106228, A364758.

Cf. A364748.

a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(5*n-4*k, n-1-k))/n);

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-4*k,n-1-k) for n > 0.

A336708 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-1)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 1, -1, 2, 1, 1, 2, 1, 0, -3, 1, 1, 3, 6, 1, 2, -1, 1, 1, 4, 14, 21, 1, 0, 11, 1, 1, 5, 25, 76, 80, 1, -5, -15, 1, 1, 6, 39, 182, 450, 322, 1, 0, -13, 1, 1, 7, 56, 355, 1447, 2818, 1347, 1, 14, 77, 1, 1, 8, 76, 611, 3532, 12175, 18352, 5798, 1, 0, -86

Offset: 0

Seiichi Manyama, Aug 01 2020

			Square array begins:
   1,  1, 1,   1,    1,     1,     1, ...
   1,  1, 1,   1,    1,     1,     1, ...
  -1,  0, 1,   2,    3,     4,     5, ...
   0, -1, 1,   6,   14,    25,    39, ...
   2,  0, 1,  21,   76,   182,   355, ...
  -3,  2, 1,  80,  450,  1447,  3532, ...
  -1,  0, 1, 322, 2818, 12175, 37206, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A007440, A090192, A000012, A106228.
Main diagonal gives A336713.
Cf. A336575, A336706, A336707, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[(-1)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-1)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+x*A)); polcoef(A, n)}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + x * A_k(x)).

Typo in name corrected by Georg Fischer, Sep 19 2023

A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704

Offset: 0

Seiichi Manyama, Aug 01 2020

			Square array begins:
    1,   1,  1,  1,   1,    1,     1, ...
    1,   1,  1,  1,   1,    1,     1, ...
   -2,  -1,  0,  1,   2,    3,     4, ...
    2,  -1, -1,  2,   8,   17,    29, ...
    4,   5,  0,  5,  36,  109,   240, ...
  -24,  -3,  2, 13, 177,  766,  2177, ...
   48, -21,  0, 36, 922, 5699, 20910, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A307969(n-1), (-1)^n * A154825(n), A090192, A246555.
Main diagonal gives A336714.
Cf. A336575, A336706, A336707, A336708.

T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-2)^(n-j)*binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1+2*x*A)); polcoef(A, n)}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k / (1 + 2 * x * A_k(x)).

A364758 G.f. A(x) satisfies A(x) = 1 + xA(x)^4 / (1 + xA(x)).

Original entry on oeis.org

1, 1, 3, 14, 76, 450, 2818, 18352, 123028, 843345, 5884227, 41650479, 298352365, 2158751879, 15754446893, 115830820439, 857147952469, 6379136387303, 47715901304501, 358529599468636, 2704884469806606, 20481615947325089, 155605509972859999, 1185779099027494848

Offset: 0

Seiichi Manyama, Aug 05 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A090192, A106228, A364759, A378919.
Cf. A001764, A219537, A364865, A365224.
Cf. A300048, A364747, A378889.

a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);

a(n, r=1, s=-1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 11 2024

a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 11 2024: (Start)
G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^5 + x*A(x)^6.
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 + x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

A215495 a(4n) = a(4n+2) = a(2n+1) = 2n + 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 7, 13, 7, 15, 9, 17, 9, 19, 11, 21, 11, 23, 13, 25, 13, 27, 15, 29, 15, 31, 17, 33, 17, 35, 19, 37, 19, 39, 21, 41, 21, 43, 23, 45, 23, 47, 25, 49, 25, 51, 27, 53, 27, 55, 29, 57, 29, 59, 31, 61, 31, 63, 33, 65, 33, 67, 35, 69, 35, 71, 37, 73, 37, 75, 39, 77, 39, 79, 41, 81, 41, 83, 43, 85, 43

Offset: 0

Paul Curtz, Aug 13 2012

A214282(n) and -A214283(n) are companions. Separately or together, they have many links with the Catalan's numbers A000108(n). Examples:
A214282(n+1) - 2*A214282(n) = -1, -1,  1,  0, -2, -5,  5,   0, -14, -42,  42,    0, -132, ....
2*A214283(n) - A214283(n+1) =  1,  0, -1, -2,  2,  0, -5, -14,  14,   0, -42, -132, 132, ....
A214282(n) + A214283(n) = 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42,... (A126120).
The companion to a(n) is b(n) = -A214283(n)/(1,1,1,1,2,2,5,5,...) = 0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, ....
a(n) - b(n) = A056594(n).
Discovered as a(n) = A214282(n+1)/A000108([n/2]). See abs(A129996(n-2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A130823, A090192.

I:=[1,1,1,3,3,5]; [n le 6 select I[n] else Self(n-2) +Self(n-4) -Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 23 2018

a[n_?EvenQ] := n/2 + Boole[Mod[n, 4] == 0]; a[n_?OddQ] := n; Table[a[n], {n, 0, 86}] (* Jean-François Alcover, Aug 14 2012 *)
LinearRecurrence[{0,1,0,1,0,-1}, {1,1,1,3,3,5}, 50] (* G. C. Greubel, Apr 23 2018 *)

x='x+O('x^30); Vec(( 1+x+2*x^3+x^4+x^5 )/( (x^2+1)*(x-1)^2*(1+x)^2 )) \\ G. C. Greubel, Apr 23 2018

a(n+3) = (A185048(n+3)=2,2,4,2,... ) + 1.
a(n+2) - a(n) = 0, 2, 2, 2. (Period 4).
a(n) = 2*a(n-4) - a(n-8).
a(2*n) = A109613(n).
a(n+1) - a(n) = 2* (-1)^n * A059169(n).
G.f. : ( 1+x+2*x^3+x^4+x^5 ) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - Jean-François Alcover, Aug 14 2012

A336727 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-k)^(n-j) * binomial(n,j) * binomial(n,j-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 0, 1, 1, 1, -3, 1, 5, 2, 1, 1, 1, -4, 5, 10, -3, 0, 1, 1, 1, -5, 11, 9, -38, -21, -5, 1, 1, 1, -6, 19, -4, -103, 28, 51, 0, 1, 1, 1, -7, 29, -35, -174, 357, 289, 41, 14, 1, 1, 1, -8, 41, -90, -203, 1176, -131, -1262, -391, 0, 1

Offset: 0

Seiichi Manyama, Aug 02 2020

			  1,  1,   1,   1,    1,    1,    1, ...
  1,  1,   1,   1,    1,    1,    1, ...
  1,  0,  -1,  -2,   -3,   -4,   -5, ...
  1, -1,  -1,   1,    5,   11,   19, ...
  1,  0,   5,  10,    9,   -4,  -35, ...
  1,  2,  -3, -38, -103, -174, -203, ...
  1,  0, -21,  28,  357, 1176, 2575, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-3 give: A000012, A090192, (-1)^n * A154825(n), A336729.
Main diagonal gives A336728.
Cf. A243631, A307884, A336708, A336709.

T[0, k_] := 1; T[n_, k_] := Sum[If[k == 0, Boole[n == j],(-k)^(n - j)] * Binomial[n, j] * Binomial[n , j - 1], {j, 1, n}] / n; Table[T[k, n- k], {n, 0, 11}, {k, 0, n}] //Flatten (* Amiram Eldar, Aug 02 2020 *)

{T(n, k) = if(n==0, 1, sum(j=1, n, (-k)^(n-j)*binomial(n, j)*binomial(n, j-1))/n)}

{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+k*x*A)); polcoef(A, n)}

{T(n, k) = sum(j=0, n, (-k)^j*(k+1)^(n-j)*binomial(n, j)*binomial(n+j, n)/(j+1))}

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x) / (1 + k * x * A_k(x)).
A_k(x) = 2/(1 - (k+1)*x + sqrt(1 + 2*(k-1)*x + ((k+1)*x)^2)).
T(n, k) = Sum_{j=0..n} (-k)^j * (k+1)^(n-j) * binomial(n,j) * binomial(n+j,n)/(j+1).
(n+1) * T(n,k) = -(k-1) * (2*n-1) * T(n-1,k) - (k+1)^2 * (n-2) * T(n-2,k) for n>1. - Seiichi Manyama, Aug 08 2020

cp's OEIS Frontend

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A336709 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.

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A364759 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 + xA(x)).

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