A049027 -id:A049027 - cp's OEIS frontend

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).

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

A110488 A number triangle based on the Catalan numbers.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jul 22 2005

Columns include A000108, A001700, A049027(n+1), A076025(n+1). Rows sums are A110489, diagonal sums are A110490.

			Rows begin
   1;
   1,  1;
   2,  2,  1;
   5,  5,  3,  1;
  14, 14, 10,  4,  1;
  42, 42, 35, 17,  5,  1;

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

T[n_, 0] := CatalanNumber[n]; T[n_, 1] := CatalanNumber[n]; T[n_, n_] := 1; T[n_, k_] := Sum[2*(j + 1)*(k - 1)^j*Binomial[2 (n - k) + 1, n - k - j]/(n - k + j + 2), {j, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 28 2017 *)

T(n, k) = Sum_{j=0..(n-k)} 2*(j+1)*(k-1)^j*C(2*(n-k)+1, n-k-j)/(n-k+j+2).
Column k has g.f. x^k*c(x)/(1-k*x*c(x)) where c(x) is the g.f. of A000108.
T(n,0) = Catalan(n), T(n,1) = Catalan(n), T(n,n) = 1. - G. C. Greubel, Aug 28 2017

A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 20, 9, 9, 1, 126, 70, 60, 30, 27, 12, 1, 462, 252, 210, 100, 105, 180, 27, 40, 54, 15, 1, 1716, 924, 756, 700, 378, 630, 300, 270, 140, 360, 108, 50, 90, 18, 1, 6435, 3432, 2772, 2520, 1225, 1386, 2268, 2100, 945, 900, 504, 1260, 600, 1080, 81

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3); the k=3 member in the family of a generalization of the multinomial number arrays M_0 = M_0(2) = A048996.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The s2(3,n,m):=A035324(n,m) numbers (generalized Pascal triangle) are obtained by summing in row n all numbers with the same part number m. In the same manner the s2(2,n,m) = binomial(n-1,m-1) = A007318(n-1,m-1) numbers are obtained from the partition array M_0 = A048996.

			[1]; [3,1]; [10,6,1]; [35,20,9,9,1]; [126,70,60,30,27,12,1]; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A049027 (row sums, also of triangle A035324).

a(n,k) = m!*Product_{j=1..n} (s2(3,j,1)^e(n,k,j))/e(n,k,j)! with s2(3,n,1) = A035324(n,1) = A001700(n-1) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A117375 Riordan array (1/(1-3x*c(x)),xc(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 3, 1, 12, 4, 1, 51, 17, 5, 1, 222, 74, 23, 6, 1, 978, 326, 104, 30, 7, 1, 4338, 1446, 468, 142, 38, 8, 1, 19323, 6441, 2103, 657, 189, 47, 9, 1, 86310, 28770, 9447, 3006, 903, 246, 57, 10, 1, 386250, 128750, 42440, 13670, 4223, 1217, 314, 68, 11, 1, 1730832

Offset: 0

Paul Barry, Mar 10 2006

Triangle factors as (1,xc(x))*(1/(1-3x),x). First row is A007854. Second row is A049027(n)-0^n. Row sums are A049027(n+1). Diagonal sums are A117376.

			Triangle begins
1,
3, 1,
12, 4, 1,
51, 17, 5, 1,
222, 74, 23, 6, 1,
978, 326, 104, 30, 7, 1,
4338, 1446, 468, 142, 38, 8, 1
Production array begins
3, 1
3, 1, 1
3, 1, 1, 1
3, 1, 1, 1, 1
3, 1, 1, 1, 1, 1
3, 1, 1, 1, 1, 1, 1
3, 1, 1, 1, 1, 1, 1, 1
3, 1, 1, 1, 1, 1, 1, 1, 1
... - _Philippe Deléham_, Mar 05 2013

Number triangle T(0,0)=1, T(n,k)=[k<=n]*sum{j=0..n, (j/(n-j))*C(2n-j,n-j)[k<=j]*3^(j-k)}

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

A370376 Number of compositions of n where there are A025174(k) sorts of part k.

Original entry on oeis.org

1, 1, 6, 39, 262, 1791, 12372, 86052, 601374, 4217151, 29648766, 208855791, 1473509736, 10408539844, 73596075552, 520797997464, 3687846866382, 26128671296127, 185209915856802, 1313356295909877, 9316374980571702, 66105343198654407, 469174119885678972

Offset: 0

Seiichi Manyama, Feb 16 2024

Cf. A005809, A025174, A049027, A370375.

my(N=30, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, binomial(3*k, k)*x^k)/3))

G.f.: 1 / (1 - 1/3 * Sum_{k>=1} binomial(3*k,k) * x^k).

a(0) = 1; a(n) = 1/3 * Sum_{k=1..n} binomial(3*k,k) * a(n-k).

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 6, 8, 6, 0, 1, 18, 30, 20, 10, 0, 1, 57, 108, 90, 40, 15, 0, 1, 186, 399, 378, 210, 70, 21, 0, 1, 622, 1488, 1596, 1008, 420, 112, 28, 0, 1, 2120, 5598, 6696, 4788, 2268, 756, 168, 36, 0, 1, 7338, 21200, 27990, 22320, 11970, 4536, 1260, 240, 45

Offset: 0

Philippe Deléham, Dec 13 2009

			Triangle begins : 1 ; 0,1 ; 1,0,1 ; 2,3,0,1 ; 6,8,6,0,1 ; 18,30,20,10,0,1 ; ...

Cf. A007318

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000957(n+1), A033321(n), A033543(n) for x = 0,1,2 respectively. Sum_{k, 0<=k<=n} T(n,k)*(-1)^(n-k)*x^k = A054341(n), A059738(n), A049027(n+1) for x = 2,3,4 respectively.

cp's OEIS Frontend

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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A110488 A number triangle based on the Catalan numbers.

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A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

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A117375 Riordan array (1/(1-3x*c(x)),xc(x)), c(x) the g.f. of A000108.

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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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Examples

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Programs

Mathematica

Formula

Extensions

A370376 Number of compositions of n where there are A025174(k) sorts of part k.

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Author

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Programs

PARI

Formula

A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

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Formula

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.