A039599 -id:A039599 - cp's OEIS frontend

A089942 Inverse binomial matrix applied to A039599.

1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 6, 15, 15, 10, 4, 1, 15, 36, 40, 29, 15, 5, 1, 36, 91, 105, 84, 49, 21, 6, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 232, 603, 750, 672, 468, 258, 111, 36, 8, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 1585, 4213, 5500

Offset: 0

Paul Barry, Nov 16 2003

Reverse of A071947 - related to lattice paths. First column is A005043.
Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Feb 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array (f(x),x*g(x)), where f(x)is the o.g.f. of A005043 and g(x)is the o.g.f. of A001006. - Philippe Deléham, Nov 22 2009
Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1+x)), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1+x)/(1+x+x^2),x/(1+x+x^2)). E.g.f. of column k is exp(x)*(Bessel_I(k,2x)-Bessel_I(k+1,2x)).
Diagonal sums are A187306.
Simultaneous equations using the first n rows solve for diagonal lengths of odd N = (2n+1) regular polygons, with constants c^0, c^1, c^2, ...; where c = 1 + 2*cos( 2*Pi/N) = sin(3*Pi/N)/sin(Pi/N) = the third longest diagonal of N>5.  By way of example, take the first 4 rows relating to the 9-gon (nonagon), N=(2*4 + 1), with c = 1 + 2*cos(2*Pi/9) = 2.5320888.... The simultaneous equations are (1,0,0,0) = 1; (0,1,0,0) = c; (1,1,1,0) = c^2, (1,3,2,1) = c^3. The answers are 1, 2.532..., 2.879..., and 1.879...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. - Gary W. Adamson, Sep 07 2011
Number of linearly independent irreducible representations of a given weight j in a tensor of given rank n. - Mikkel N. Schmidt, Aug 20 2025

			Triangle begins
   1,
   0,   1,
   1,   1,   1,
   1,   3,   2,   1,
   3,   6,   6,   3,   1,
   6,  15,  15,  10,   4,  1,
  15,  36,  40,  29,  15,  5,  1,
  36,  91, 105,  84,  49, 21,  6, 1,
  91, 232, 280, 238, 154, 76, 28, 7, 1
Production matrix is
  0, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry and Aoife Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
J. A. R. Coope, R. F. Snider, and F. R. McCourt, Irreducible Cartesian Tensors, The Journal of Chemical Physics, vol 43, no 7, 1965.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Toufik Mansour and Mark Shattuck, Enumeration of Catalan and smooth words according to capacity, Integers (2025) Vol. 25, Art. No. A5. See p. 29.
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths, Eur. J. Comb. (2014) Vol. 39, 157-169. See Table 2.2.

Row sums give A002426 (central trinomial coefficients).

T:= (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)-GegenbauerC(n-k-1,-n+1,-1/2)): for n from 1 to 9 do seq(T(n,k), k=1..n) od; # Peter Luschny, May 12 2016
# Or by recurrence:
T := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0
elif k = 0 then T(n-1, 1) else T(n-1, k-1) + T(n-1, k) + T(n-1, k+1) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 25 2021

T[n_, k_] := GegenbauerC[n - k, -n + 1, -1/2] - GegenbauerC[n - k - 1, -n + 1, -1/2]; Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

G.f.: (1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A005043(m+n). - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*(2k+1) = 3^n. - Philippe Deléham, Mar 22 2007
Sum_{k=0..n} T(n,k)*2^k = A112657(n). - Philippe Deléham, Apr 01 2007
T(n,2k) + T(n,2k+1) = A109195(n,k). - Philippe Deléham, Nov 11 2008
T(n,k) = GegenbauerC(n-k,-n+1,-1/2) - GegenbauerC(n-k-1,-n+1,-1/2) for 1 <= k <= n. - Peter Luschny, May 12 2016

Edited by Emeric Deutsch, Mar 04 2004

A129818 Riordan array (1/(1+x), x/(1+x)^2), inverse array is A039599.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -1, 6, -5, 1, 1, -10, 15, -7, 1, -1, 15, -35, 28, -9, 1, 1, -21, 70, -84, 45, -11, 1, -1, 28, -126, 210, -165, 66, -13, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1, -1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1

Offset: 0

Philippe Deléham, Jun 09 2007

This sequence is up to sign the same as A129818. - T. D. Noe, Sep 30 2011
Row sums: A057078. - Philippe Deléham, Jun 11 2007
Subtriangle of the triangle given by (0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012
This triangle provides the coefficients of powers of x^2 for the even-indexed Chebyshev S polynomials (see A049310): S(2*n,x) = Sum_{k=0..n} T(n,k)*x^(2*k), n >= 0. - Wolfdieter Lang, Dec 17 2012
If L(x^n) := C(n) = A000108(n) (Catalan numbers), then the polynomials P_n(x) := Sum_{k=0..n} T(n,k)*x^k are orthogonal with respect to the inner product given by (f(x),g(x)) := L(f(x)*g(x)). - Michael Somos, Jan 03 2019

			Triangle T(n,k) begins:
  n\k  0   1    2     3     4     5    6    7    8   9 10 ...
   0:  1
   1: -1   1
   2:  1  -3    1
   3: -1   6   -5     1
   4:  1 -10   15    -7     1
   5: -1  15  -35    28    -9     1
   6:  1 -21   70   -84    45   -11    1
   7: -1  28 -126   210  -165    66  -13    1
   8:  1 -36  210  -462   495  -286   91  -15    1
   9: -1  45 -330   924 -1287  1001 -455  120  -17   1
  10:  1 -55  495 -1716  3003 -3003 1820 -680  153 -19  1
  ... Reformatted by _Wolfdieter Lang_, Dec 17 2012
Recurrence from the A-sequence A115141:
15 = T(4,2) = 1*6 + (-2)*(-5) + (-1)*1.
(0, -1, 0, -1, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, ...) begins:
  1
  0,  1
  0, -1,   1
  0,  1,  -3,   1
  0, -1,   6,  -5,  1
  0,  1, -10,  15, -7,  1
  0, -1,  15, -35, 28, -9, 1. - _Philippe Deléham_, Mar 19 2012
Row polynomial for n=3 in terms of x^2: S(6,x) = -1 + 6*x^2 -5*x^4 + 1*x^6, with Chebyshev's S polynomial. See a comment above. - _Wolfdieter Lang_, Dec 17 2012
Boas-Buck type recurrence: -35 = T(5,2) = (5/3)*(-1*1 +1*(-5) - 1*15) = -3*7 = -35. - _Wolfdieter Lang_, Jun 03 2020

Vincenzo Librandi, Rows n = 1..101, flattened
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry and Aoife Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, example 15.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Cf. A039599, A085478, A123970, A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare((1 - sqrt(1 - 4*x))/(2*x), 10):
LinearAlgebra[MatrixInverse](%); # Peter Luschny, Jan 04 2019

max = 10; Flatten[ CoefficientList[#, y] & /@ CoefficientList[ Series[ (1 + x)/(1 + (2 - y)*x + x^2), {x, 0, max}], x]] (* Jean-François Alcover, Sep 29 2011, after Wolfdieter Lang *)

@CachedFunction
def A129818(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A129818(n-1,k) if n==1 else 2*A129818(n-1,k)
    return A129818(n-1,k-1) - A129818(n-2,k) - h
for n in (0..9): [A129818(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,k) = (-1)^(n-k)*A085478(n,k) = (-1)^(n-k)*binomial(n+k,2*k).
Sum_{k=0..n} T(n,k)*A000531(k) = n^2, with A000531(0)=0. - Philippe Deléham, Jun 11 2007
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A057078(n), A057077(n), A057079(n), A005408(n), A002878(n), A001834(n), A030221(n), A002315(n), A033890(n), A057080(n), A057081(n), A054320(n), A097783(n), A077416(n), A126866(n), A028230(n+1) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, respectively. - Philippe Deléham, Nov 19 2009
O.g.f.: (1+x)/(1+(2-y)*x+x^2). - Wolfdieter Lang, Dec 15 2010
O.g.f. column k with leading zeros (Riordan array, see NAME): (1/(1+x))*(x/(1+x)^2)^k, k >= 0. - Wolfdieter Lang, Dec 15 2010
From Wolfdieter Lang, Dec 20 2010: (Start)
Recurrences from the Z- and A-sequences for Riordan arrays. See the W. Lang link under A006232 for details and references.
T(n,0) = -1*T(n-1,0), n >= 1, from the o.g.f. -1 for the Z-sequence (trivial result).
T(n,k) = Sum_{j=0..n-k} A(j)*T(n-1,k-1+j), n >= k >= 1, with A(j):= A115141(j) = [1,-2,-1,-2,-5,-14,...], j >= 0 (o.g.f. 1/c(x)^2 with the A000108 (Catalan) o.g.f. c(x)). (End)
T(n,k) = (-1)^n*A123970(n,k). - Philippe Deléham, Feb 18 2012
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 19 2012
A039599(m,n) = Sum_{k=0..n} T(n,k) * C(k+m) where C(n) are the Catalan numbers. - Michael Somos, Jan 03 2019
Equals the matrix inverse of the Riordan square (cf. A321620) of the Catalan numbers. - Peter Luschny, Jan 04 2019
Boas-Buck type recurrence for column k >= 0 (see Aug 10 2017 comment in A046521 with references): T(n,k) = ((1 + 2*k)/(n - k))*Sum_{j = k..n-1} (-1)^(n-j)*T(j,k), with input T(n,n) = 1, and T(n,k) = 0 for n < k. - Wolfdieter Lang, Jun 03 2020

A127872 Triangle formed by reading A039599 mod 2.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1

Offset: 0

Philippe Deléham, Apr 05 2007

Also triangle formed by reading triangles A061554, A106180, A110519, A124574, A124576, A126953, A127543 modulo 2.

			Triangle begins:
1;
1, 1;
0, 1, 1;
1, 1, 1, 1;
0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 1, 1, 1, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1;
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1;
0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A061554, A106180, A110519, A124574, A124576, A126953, A127543.

Cf. A000007, A036987, A001316, A062878, A050614, A000045, A001519

T[0, 0] := 1; T[n_, k_] := Binomial[2*n - 1, n - k] - Binomial[2*n - 1, n - k - 2]; Table[Mod[T[n, k], 2], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Apr 18 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A036987(n), A001316(n), A062878(n) for x=-1,0,1,2 respectively.

Sum_{k=0..n} T(n,k)*Fibonacci(2*k+1) = A050614(n), see A000045 and A001519. - Philippe Deléham, Aug 30 2007

A125275 Eigensequence of triangle A039599: a(n) = Sum_{k=0..n-1} A039599(n-1,k)*a(k) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 7, 31, 162, 968, 6481, 47893, 386098, 3364562, 31460324, 313743665, 3320211313, 37124987124, 436985496790, 5397178181290, 69748452377058, 940762812167126, 13213888481979449, 192891251215160017

Offset: 0

Paul D. Hanna, Nov 26 2006

nonn

Starting with offset 1, these are the row sums of triangle A147294. - Gary W. Adamson, Nov 05 2008

			a(3) = 2*(1) + 3*(1) + 1*(2) = 7;
a(4) = 5*(1) + 9*(1) + 5*(2) + 1*(7) = 31;
a(5) = 14*(1) + 28*(1) + 20*(2) + 7*(7) + 1*(31) = 162.
Triangle A039599(n,k) = C(2*n+1, n-k)*(2*k+1)/(2*n+1) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   1,  1;
   2,  3,  1;
   5,  9,  5,  1;
  14, 28, 20,  7, 1;
  42, 90, 75, 35, 9, 1;
  ...
where the g.f. of column k is G000108(x)^(2*k+1)
and G000108(x) = (1 - sqrt(1 - 4*x))/(2*x) is the Catalan g.f. function.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A039599, A125276 (variant), A147294.

A125275=ConstantArray[0,20]; A125275[[1]]=1; Do[A125275[[n]]=Binomial[2*n-1,n-1]/(2*n-1)+Sum[A125275[[k]]*Binomial[2*n-1,n-k-1]*(2*k+1)/(2*n-1),{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125275}] (* Vaclav Kotesovec, Dec 09 2013 *)

a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(2*n-1, n-k-1)*(2*k+1)/(2*n-1)))

a(n) = Sum_{k=0..n-1} a(k) * C(2*n-1, n-k-1) * (2*k + 1)/(2*n - 1) for n > 0 with a(0) = 1.

A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 6, 2, 1, 0, 0, 10, 8, 2, 1, 0, 0, 20, 16, 12, 2, 1, 0, 0, 35, 47, 25, 17, 2, 1, 0, 0, 70, 94, 97, 36, 23, 2, 1, 0, 0, 126, 244, 204, 179, 49, 30, 2, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 07 2009

Diagonal sums: A089324.

Equal to A092107*B^(-1) = A092107*A130595 as lower triangular arrays. - Philippe Deléham, Dec 10 2009

			Triangle begins:
   1;
   1, 0;
   2, 0, 0;
   3, 1, 0, 0;
   6, 2, 1, 0, 0;
  10, 8, 2, 1, 0, 0;
  ...

Cf. A000108, A001006, A001405, A033321, A089324.

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = 0, 1, 2, 3 respectively.

A152038 Convolution of A039599 with itself .

Original entry on oeis.org

1, 2, 3, 6, 11, 12, 22, 42, 51, 56, 107, 170, 201, 272, 385, 476, 653, 1042, 1433, 1590, 1602, 2386, 4280, 5644, 5508, 6110, 9217, 13324, 17315, 19512, 19690, 25990, 43918, 59982, 61931, 57374

Offset: 0

Philippe Deléham, Nov 20 2008

nonn

Cf. A039599

lista(nn) = {v = vector((nn+1)*(nn+2)/2); iv = 0; for (n=0, nn, for (k=0, n, v[iv++] = (2*k+1)*binomial(2*n, n-k)/(n+k+1););); n = #v; for (i=1, n, print1(sum(k=1, i, v[k]*v[i-k+1]), ", "););} \\ Michel Marcus, Jun 02 2013

More terms from Michel Marcus, Jun 02 2013

A160052 Triangle : Lodumo_2 applied to each row of triangle in A039599 .

Original entry on oeis.org

1, 1, 3, 0, 1, 3, 1, 3, 5, 7, 0, 2, 4, 1, 3, 0, 2, 1, 3, 5, 7, 0, 1, 3, 2, 4, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 6, 8, 10, 12, 1, 3, 0, 2, 4, 6, 8, 10, 1, 3, 5, 7, 0, 2, 4, 6, 8, 1, 3, 10, 12, 5, 7, 0, 2, 4, 6, 1, 3, 5, 7, 9, 11, 13, 15, 0, 2, 4, 1, 3, 6, 8, 10, 12, 14, 16, 5, 7

Offset: 0

Philippe Deléham, Apr 30 2009

Row sums: A160020.

			Triangle begins : 1 ; 1,3 ; 0,1,3 ; 1,3,5,7 ; 0,2,4,1,3 ; 0,2,1,3,5,7,

A169589 A number triangle with repeated columns of triangle in A039599.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 2, 3, 1, 14, 5, 9, 3, 1, 42, 14, 28, 9, 5, 1, 132, 42, 90, 28, 20, 5, 1, 429, 132, 297, 90, 75, 20, 7, 1, 1430, 429, 1001, 297, 275, 75, 35, 7, 1, 4862, 1430, 3432, 1001, 1001, 275, 154, 35, 9, 1, 16796, 4862, 11934, 3432, 3640, 1001, 637, 154, 54, 9, 1

Offset: 0

Philippe Deléham, Dec 02 2009

Row sums : A135339.

			Triangle begins : 1 ; 1,1 ; 2,1,1 ; 5,2,3,1 ; 14,5,9,3,1 ; 42,14,28,20,5,1 ; ...

Cf. A000012, A109613, A000108

Sum_{k, 0<=k<=n} T(n,k)= A135339(n+1). T(n,0)= A000108(n).

Corrected by Philippe Deléham, Dec 04 2009

A171388 Expansion of the first column of triangle T_(2,x), T_(x,y) defined in A039599; T_(2,0)= A126075, T_(2,1)= A038622, T_(2,2)= A039598, T_(2,3)= A124733, T_(2,4)= A124575.

Original entry on oeis.org

1, 2, 0, 5, 0, 0, 12, 1, 0, 0, 30, 4, 1, 0, 0, 74, 17, 4, 1, 0, 0, 185, 56, 21, 4, 1, 0, 0, 460, 185, 74, 26, 4, 1, 0, 0

Offset: 0

Philippe Deléham, Dec 07 2009

			Triangle begins:
   1;
   2,  0;
   5,  0, 0;
  12,  1, 0, 0;
  30,  4, 1, 0, 0;
  74, 17, 4, 1, 0, 0;
  ...

Cf. A000108, A171368, A171380.

Sum_{k=0..n} T(n,k)*x^k = A054341(n), A005773(n+1), A000108(n+1), A007317(n), A033543(n) for x = 0, 1, 2, 3, 4 respectively.

A382403 a(n) = Sum_{k=0..n} A039599(n,k)^3.

Original entry on oeis.org

1, 2, 36, 980, 33040, 1268568, 53105976, 2364239592, 110206067400, 5323547715200, 264576141331216, 13458185494436592, 697931136204820336, 36789784967375728400, 1966572261077797609200, 106400946932857148590800, 5817987630644593688220600, 321105713814359742307398480

Offset: 0

Seiichi Manyama, Mar 24 2025

nonn

Let b_k(n) = Sum_{j=0..n} A039599(n,j)^k. b_1(n) = binomial(2*n,n) = A000984(n) and b_2(n) = binomial(4*n,2*n)/(2*n+1) = A048990(n).

Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Cf. A000984, A048990.

Cf. A039599, A112029.

a039599(n, k) = (2*k+1)/(n+k+1)*binomial(2*n, n-k);
a(n) = sum(k=0, n, a039599(n, k)^3);

a(n) = binomial(2*n,n) * (4 * binomial(2*n,n)^2 - 3 * A112029(n)).

cp's OEIS Frontend

A089942 Inverse binomial matrix applied to A039599.

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A129818 Riordan array (1/(1+x), x/(1+x)^2), inverse array is A039599.

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A127872 Triangle formed by reading A039599 mod 2.

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A125275 Eigensequence of triangle A039599: a(n) = Sum_{k=0..n-1} A039599(n-1,k)*a(k) for n > 0 with a(0) = 1.

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A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

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A152038 Convolution of A039599 with itself .

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A160052 Triangle : Lodumo_2 applied to each row of triangle in A039599 .

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A169589 A number triangle with repeated columns of triangle in A039599.

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