A050144 -id:A050144 - cp's OEIS frontend

A050166 Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.

1, 1, 2, 1, 4, 5, 1, 6, 14, 14, 1, 8, 27, 48, 42, 1, 10, 44, 110, 165, 132, 1, 12, 65, 208, 429, 572, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 16, 119, 544, 1700, 3808, 6188, 7072, 4862, 1, 18, 152, 798, 2907, 7752, 15504, 23256, 25194, 16796

Offset: 0

Clark Kimberling

Sometimes called Catalan's triangle, although this term is usually reserved for several other triangles!
T is a mirror image of the array in A039598.
Given (1) = row 0, then the sum of terms with alternating signs in row r of A050166 = (-1)^r * A000108(n); where A000108 = 1, 1, 2, 5, 14, 42, ...the Catalan numbers. - Herb Conn
The diagonals of this triangle are self-convolutions of the main diagonal A000108(n+1): 1, 2, 5, 14, 42, 132, 429, ... - Philippe Deléham, May 25 2005
The multiplicities of the eigenvalues of the middle cubes are related to this triangle. The middle cube in Q_3 has eigenvalues -2, -1, 1, 2 with multiplicities 1, 2, 2, 1. The middle cube in Q_5 has eigenvalues -3, -2, -1, 1, 2, 3 with multiplicities 1, 4, 5, 5, 4, 1. The middle cube in Q_7 has eigenvalues -4, -3, -2, -1, 1, 2, 3, 4 with multiplicities 1, 6, 14, 14, 14, 14, 6, 1, etc. - Ke Qiu, Apr 05 2019

			Triangle begins:
  1;
  1,  2;
  1,  4,  5;
  1,  6, 14, 14;
  1,  8, 27, 48, 42;
  ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Y. Jiang, K. Qiu, R. Qiu, and J. Shen, On the spectrum of the middle-cube, Congressus Numerantium, 195 (2009), 195-204.
A. Nkwanta, Lattice paths and RNA secondary structures, in: Nathaniel Dean, African Americans in Mathematics, AMS and DIMACS, 1997, ISBN 978-0-8218-0678-4, pp. 137-147.

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
L. W. Shapiro, W.-J. Woan and S. Getu, Runs, slides and moments, SIAM J. Alg. Discrete Methods, 4 (1983), 459-466.

Mirror image of A039598.

Flat(List([0..10], n-> List([0..n], k-> 2*(n-k+1)* Binomial(2*n+1, k)/(2*n-k+2) ))); # G. C. Greubel, Apr 05 2019

[[2*(n-k+1)*Binomial(2*n+1,k)/(2*n-k+2): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019

Table[2*Binomial[2n+1, k]*(n-k+1)/(2*n-k+2), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)

{T(n,k) = 2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2)}; \\ G. C. Greubel, Apr 05 2019

[[2*(n-k+1)*binomial(2*n+1,k)/(2*n-k+2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019

From Henry Bottomley, Sep 24 2001: (Start)
T(n, k) = C(2n+1, k)*2*(n-k+1)/(2n-k+2) = A039598(n, n-k)
T(n, k) = T(n-1, k) + 2*T(n-1, k-1) + T(n-1, k-2), with T(0, 0) = 1 and T(n, k) = 0 if n < 0 or n < k. (End)
Sum_{0<=k<=n} T(n,k)*x^k = A000012(n), A001700(n), A194723(n+1), A194724(n+1), A194725(n+1), A194726(n+1), A195727(n+1), A194728(n+1), A194729(n+1), A194730(n+1) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Nov 03 2011

More terms from Larry Reeves (larryr(AT)acm.org), Mar 14 2001

A050165 Triangle read by rows: T(n,k) = M(2n+1,k,-1), 0 <= k <= n, n >= 0, array M as in A050144.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 28, 14, 1, 9, 35, 75, 90, 42, 1, 11, 54, 154, 275, 297, 132, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 15, 104, 440, 1260, 2548, 3640, 3432, 1430, 1, 17, 135, 663, 2244, 5508, 9996, 13260, 11934

Offset: 0

Clark Kimberling

T is a mirror image of the array in A039599.

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,   5;
  1,  7, 20,  28,  14;
  1,  9, 35,  75,  90,  42;
  1, 11, 54, 154, 275, 297, 132;

M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015. See Section 4.

Cf. A039599, A084938.

Triangle T(n, k) read by rows; given by [1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938. T(n, k) = C(2n, k)*(2n-2k+1)/(2n-k+1). - Philippe Deléham, Dec 07 2003
Sum_{k=0..min(m, n)} T(m, m-k)*T(n, n-k) = A000108(m+n); A000108: Catalan numbers. - Philippe Deléham, Dec 30 2003
T(n, k) = 0 if n < k, T(n, n)= A000108(n) and for n > k: T(n, k) = Sum_{j=0..k} T(n-1-j, k-j)*A000108(j+1). - Philippe Deléham, Feb 03 2004
T(n,k)= Sum_{j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k). - Philippe Deléham, May 05 2007
T(2n,n) = A126596(n). - Philippe Deléham, Nov 23 2011

A050153 T(n,k)=M(2n+1,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 3, 1, 1, 9, 5, 6, 1, 28, 20, 27, 8, 1, 90, 75, 110, 44, 10, 1, 297, 275, 429, 208, 65, 12, 1, 1001, 1001, 1638, 910, 350, 90, 14, 1, 3432, 3640, 6188, 3808, 1700, 544, 119, 16, 1

Offset: 0

Clark Kimberling

			Rows: {0}; {1,0}; {3,1,1}; ...

A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 5, 4, 5, 1, 14, 14, 20, 7, 1, 42, 48, 75, 35, 9, 1, 132, 165, 275, 154, 54, 11, 1, 429, 572, 1001, 637, 273, 77, 13, 1, 1430, 2002, 3640, 2548, 1260, 440, 104, 15, 1, 4862, 7072, 13260, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

Clark Kimberling

First 7 columns of T are A000108, A002057, A000344, A000588, A001392, A000589, A000590.

			Rows: {0}; {1,0}; {2,1,1}; ...

A050154 T(n,k)=M(2n+2,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

0, 1, 0, 4, 1, 1, 14, 6, 7, 1, 48, 27, 35, 9, 1, 165, 110, 154, 54, 11, 1, 572, 429, 637, 273, 77, 13, 1, 2002, 1638, 2548, 1260, 440, 104, 15, 1, 7072, 6188, 9996, 5508, 2244, 663, 135, 17, 1

Offset: 0

Clark Kimberling

			Rows: {0}; {1,0}; {4,1,1}; ...

A050156 T(n,k)=M(2n+3,n+3,k+3), 0<=k<=n, n >= 0, array M as in A050144.

Original entry on oeis.org

1, 4, 1, 14, 6, 1, 48, 27, 8, 1, 165, 110, 44, 10, 1, 572, 429, 208, 65, 12, 1, 2002, 1638, 910, 350, 90, 14, 1, 7072, 6188, 3808, 1700, 544, 119, 16, 1, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1

Offset: 0

Clark Kimberling

			Rows: {1}; {4,1}; {14,6,1}; ...

A050167 T(n,k)=M(n,k,f(n,k)), 0<=k<=n, n >= 0, array M as in A050144 and f(n,k)=least number t for which M(n,k,t) is not 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 3, 2, 5, 4, 1, 1, 4, 5, 5, 9, 5, 1, 1, 5, 9, 5, 14, 14, 6, 1, 1, 6, 14, 14, 14, 28, 20, 7, 1, 1, 7, 20, 28, 14, 42, 48, 27, 8, 1, 1, 8, 27, 48, 42, 42, 90, 75, 35, 9, 1, 1, 9, 35, 75, 90, 42, 132, 165, 110, 44

Offset: 0

Clark Kimberling

f(n,k)=-1 if 0<=k<=[ (n-1)/2 ], f(n,k)=2k-n if [ (n+1)/2 ]+1<=k<=n.

			Rows: {1}; {1,1}; {1,1,1}; {1,1,2,1}, ...

A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

Original entry on oeis.org

1, 3, 1, 9, 5, 1, 28, 20, 7, 1, 90, 75, 35, 9, 1, 297, 275, 154, 54, 11, 1, 1001, 1001, 637, 273, 77, 13, 1, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1, 41990, 48450, 38760, 23256, 10659, 3705, 950, 170, 19, 1

Offset: 1

Clark Kimberling

T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. Zoran Sunic reference) . - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+k+1, n-k-1). - Emeric Deutsch, May 30 2004
Riordan array (c(x)^3,xc(x)^2) where c(x) is the g.f. of A000108. Inverse array is A109954. - Paul Barry, Jul 06 2005

			    1;
    3,   1;
    9,   5,   1;
   28,  20,   7,  1;
   90,  75,  35,  9,  1;
  297, 275, 154, 54, 11, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
R. K. Guy, Catwalks, Sansteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), #00.1.6
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math 14 (1957), 405-414.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
_Zoran Sunic_, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, 10 (2003) #N5.

First columns: A000245, A000344, A000588, A001392, A000589, A000590.

Cf. A000108, A001791 (row sums), A050144.

T:= (n, k)->  (2*k+3)*binomial(2*n, n-k-1)/(n+k+2):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 19 2013

T[n_, k_] :=  (2*k + 3)*Binomial[2*n, n - k - 1]/(n + k + 2);
Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 21 2016 *)

Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).
G.f. of column k: x^(k+1)*C^(2*k+3) where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Feb 03 2004
T(n, k) = A039599(n, k+1) = A009766(n+k+1, n-k-1) = A033184(n+k+2, 2k+3) . - Philippe Deléham, May 28 2005
Sum_{k>= 0} T(m, k)*T(n, k) = A000108(m+n) - A000108(m)*A000108(n). - Philippe Deléham, May 28 2005
T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - Paul Barry, Jul 06 2005

Edited by Philippe Deléham, May 22 2005

A050159 T(n,k) = S(2n-1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 5, 9, 10, 5, 14, 28, 34, 35, 14, 42, 90, 117, 125, 126, 42, 132, 297, 407, 451, 461, 462, 132, 429, 1001, 1430, 1638, 1703, 1715, 1716, 429, 1430, 3432, 5070, 5980, 6330, 6420, 6434, 6435, 1430, 4862, 11934, 18122

Offset: 0

Clark Kimberling

			Rows: {0}; {1,1}; {1,2,3}; ...

Cf. A050144, A050157.

T(n, k) = Sum_{t(n, j): 0<=j<=k}, array t as in A050144.

cp's OEIS Frontend

A050166 Triangle T(n,k) = M(2n,k,-1), with 0 <= k <= n, n >= 0, and array M is defined in A050144.

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A050165 Triangle read by rows: T(n,k) = M(2n+1,k,-1), 0 <= k <= n, n >= 0, array M as in A050144.

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A050153 T(n,k)=M(2n+1,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

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A050145 T(n,k)=M(2n,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

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A050154 T(n,k)=M(2n+2,n-1,k-1), 0<=k<=n, n >= 0, array M as in A050144.

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A050156 T(n,k)=M(2n+3,n+3,k+3), 0<=k<=n, n >= 0, array M as in A050144.

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A050167 T(n,k)=M(n,k,f(n,k)), 0<=k<=n, n >= 0, array M as in A050144 and f(n,k)=least number t for which M(n,k,t) is not 0.

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A050155 Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).

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A050159 T(n,k) = S(2n-1,n-1,k-1), 0<=k<=n, n >= 0, array S as in A050157.

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