A067347 -id:A067347 - cp's OEIS frontend

A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

1, 3, 12, 51, 222, 978, 4338, 19323, 86310, 386250, 1730832, 7763550, 34847796, 156503064, 703149438, 3160160811, 14206181382, 63874779714, 287242041528, 1291872728826, 5810776384932, 26138647551564, 117587214581508

Offset: 0

Martin Klazar

Chains in rooted plane trees on n nodes.
The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry, Jan 20 2007
Binomial transform of A112657. - Philippe Deléham, Nov 25 2007
Row sums of the Riordan matrix (1/sqrt(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A116395). - Emanuele Munarini, Apr 26 2011
Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 3*x*C(x)) (mod 2). - Peter Bala, Jul 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Martin Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level, arXiv:math/0405597 [math.CO], 2004.
Index entries for sequences related to rooted trees

Cf. A000108, A000984, A076035, A076036, A067347, A116395, A126694.

CoefficientList[Series[(1+3Sqrt[1-4x])/(4-18x),{x,0,25}],x] (* Emanuele Munarini, Apr 26 2011 *)
nm = 25; t = NestList[Append[Accumulate[#], 3 Total[#]] &, {1}, nm];
Table[t[[n, n]], {n, nm}] (*similar to generating Catalan's triangle A009766*)
(* Li Han, Oct 23 2020 *)

makelist(kron_delta(n,0)+sum(binomial(2*n-k,n-k)*(k*3^k)/(2*n-k),k,1,n),n,0,12); /* Emanuele Munarini, Apr 26 2011 */

a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1) + A067336(n-1) = A067347(3, n-1). - Henry Bottomley, Jan 16 2002
a(n) = Sum_{k>=0} A106566(n, k)*3^k. - Philippe Deléham, Aug 11 2005
The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - Philippe Deléham, Nov 26 2006
a(n) = Sum_{k = 0..n} C(2n,n-k)(2k+1)2^k/(n+k+1). - Paul Barry, Jan 20 2007
a(n) = Sum_{k = 0..n} A039599(n,k)*2^k. - Philippe Deléham, Sep 08 2007
a(n) = Sum_{k = 0..n} A116395(n,k). - Vladimir Kruchinin, Mar 09 2011
From Emanuele Munarini, Apr 26 2011 (Start)
a(n) = Sum_{k = 1..n} C(2*n-k,n-k)*(k*3^k)/(2*n-k), for n>0.
a(n) = (1/4)*(9/2)^n-3*Sum_{k=0..n} C(2*k,k)/(2k-1)*(9/2)^(n-k).
D-finite with recurrence: 2*(n+2)*a(n+2)-(17*n+22)*a(n+1)+18*(2*n+1)*a(n)=0. (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = upper left term in M^n, M = the infinite square production matrix:
  3, 3, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)

More terms from Henry Bottomley, Jan 16 2002

A067345 Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1))/(n-1) with a(n,1)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 10, 4, 1, 42, 35, 17, 5, 1, 132, 126, 74, 26, 6, 1, 429, 462, 326, 137, 37, 7, 1, 1430, 1716, 1446, 726, 230, 50, 8, 1, 4862, 6435, 6441, 3858, 1434, 359, 65, 9, 1, 16796, 24310, 28770, 20532, 8952, 2582, 530, 82, 10, 1, 58786, 92378, 128750

Offset: 1

Henry Bottomley, Jan 16 2002

Also table given by Sum_{k, 0<=k<=n}A039598(n,k)*x^k ; table begins : x=0 : 1, 2, 5, 42, 132, ...(see A000108); x=1 : 1, 3, 10, 35, 126, ...(see A001700); x=2 : 1, 4, 17, 74, 326, ...(see A049027); x=3 : 1, 5, 26, 137, 726, ...(see A075025); x=4 : 1, 6, 37, 230, 1434, ...(see A075026); x=5 : 1, 7, 50, 359, 2582, ... - Philippe Deléham, Mar 21 2007

Rows include A000108, A001700, A049027. Columns essentially include A000012, A000027, A002522.

T(n, k) =A067346(n, k)/(n-1) =A067347(n, k)/n

A067346 Square array read by antidiagonals: T(n,k)=T(n,k-1)*n^2/(n-1)-Catalan(k-1) with a(n,1)=n-1 and a(1,k)=0 where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 0, 10, 8, 3, 0, 35, 34, 15, 4, 0, 126, 148, 78, 24, 5, 0, 462, 652, 411, 148, 35, 6, 0, 1716, 2892, 2178, 920, 250, 48, 7, 0, 6435, 12882, 11574, 5736, 1795, 390, 63, 8, 0, 24310, 57540, 61596, 35808, 12910, 3180, 574, 80, 9, 0, 92378, 257500

Offset: 1

Henry Bottomley, Jan 16 2002

Rows include A000004, A001700, A067336.

T(n, k) =A067345(n, k)*(n-1) =A067347(n, k)*(1-1/n)

A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)xC)/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 5, 1, 1, 4, 10, 14, 14, 1, 1, 5, 17, 35, 42, 42, 1, 1, 6, 26, 74, 126, 132, 132, 1, 1, 7, 37, 137, 326, 462, 429, 429, 1, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 1, 10, 82

Offset: 0

N. J. A. Sloane, Oct 29 2002

			Array begins
1 1 1 2 5 14 42 ... (n=0)
1 1 2 5 14 42 132 ... (n=1)
1 1 3 10 35 126 ... (n=2)
1 1 4 17 74 326 ...

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076038, A067347.

C(x)=(1/2-1/2*(1-4*x)^(1/2))/x; D(x)=(1-(m-1)*x*C(x))/(1-m*x*C(x)); for(i=0,15, forstep(m=i,0,-1,print1(polcoeff(D(x),i-m),","));print()) (Klasen)

More terms from Lambert Klasen (lambert.klasen(AT)gmx.de), Jan 12 2005

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 10, 14, 14, 1, 5, 17, 35, 42, 42, 1, 6, 26, 74, 126, 132, 132, 1, 7, 37, 137, 326, 462, 429, 429, 1, 8, 50, 230, 726, 1446, 1716, 1430, 1430, 1, 9, 65, 359, 1434, 3858, 6441, 6435, 4862, 4862, 1, 10, 82, 530, 2582, 8952, 20532, 28770, 24310, 16796, 16796

Offset: 0

N. J. A. Sloane, Oct 29 2002

			Array begins as:
  1 1  2  5  14  42 ... (n=0)
  1 2  5 14  42 132 ... (n=1)
  1 3 10 35 126 ... (n=2)
  1 4 17 74 326 ...
  ...

Rows give A000108, A000108, A001700, A049027, A076025, A076026. Cf. A076037, A067347.

Cf. A059718.

Unprotect[Power]; Power[0,0]=1; Protect[Power]; A[n_, m_]:= 1/(m+1)*Sum[Binomial[2*m-k, m]*(k+1)*(n-m)^k,{k,0,m}]; Table[A[n,m],{n,0,10},{m,0,n}]//Flatten (* Stefano Spezia, Sep 01 2025 *)

A(n, m) = 1/(m+1)*Sum_{k=0..m} binomial(2*m-k, m)*(k+1)*(n-m)^k, m=0..n.

More terms from Vladeta Jovovic, Jul 18 2003

a(63)-a(65) from Stefano Spezia, Sep 01 2025

cp's OEIS Frontend

A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

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A067345 Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1))/(n-1) with a(n,1)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

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A067346 Square array read by antidiagonals: T(n,k)=T(n,k-1)*n^2/(n-1)-Catalan(k-1) with a(n,1)=n-1 and a(1,k)=0 where Catalan(k)=C(2k,k)/(k+1)=A000108(k).

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A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)*x*C)/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-n*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

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A007854 Expansion of 1/(1 - 3xC(x)), where C(x) = (1 - sqrt(1 - 4x))/(2x) = g.f. for the Catalan numbers A000108.

A076037 Square array read by antidiagonals in which row n has g.f. (1-(n-1)xC)/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.

A076038 Square array read by ascending antidiagonals in which row n has g.f. C/(1-nxC) where C = (1/2-1/2(1-4x)^(1/2))/x = g.f. for Catalan numbers A000108.