A001392 -id:A001392 - cp's OEIS frontend

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).

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

A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 14, 20, 1, 42, 75, 9, 132, 275, 54, 1, 429, 1001, 273, 13, 1430, 3640, 1260, 104, 1, 4862, 13260, 5508, 663, 17, 16796, 48450, 23256, 3705, 170, 1, 58786, 177650, 95931, 19019, 1309, 21, 208012, 653752, 389367, 92092, 8602, 252, 1

Offset: 0

Paul D. Hanna, May 10 2006

Closely related to triangle A118919.
Row n contains 1+floor(n/2) terms.
From Peter Bala, Mar 20 2009: (Start)
Combinatorial interpretations of T(n,k):
1) The number of standard tableaux of shape (n-2*k,n+2*k).
2) The entries in column k are (with an offset of 2*k) the number of n-th generation vertices in the tree of sequences with unit increase labeled by 4*k. See [Sunik, Theorem 4]. (End)

			Triangle begins:
     1;
     1;
     2,     1;
     5,     5;
    14,    20,    1;
    42,    75,    9;
   132,   275,   54,   1;
   429,  1001,  273,  13;
  1430,  3640, 1260, 104,  1;
  4862, 13260, 5508, 663, 17; ...

Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003). - _Peter Bala_, Mar 20 2009

Cf. A119244 (eigenvector), A088218, A000108, A000344, A001392; A118919 (variant), A158483; A002057, A002894.

f1 = (1-Sqrt[1-4*x])/(2*x);
DeleteCases[CoefficientList[Normal@Series[f1/(1 - x^2*y*f1^4),{x,0,10},{y,0,5}],{x,y}],0,Infinity]//TableForm  (* Bradley Klee, Feb 26 2018 *)
Table[(1+4*k)/(n+1+2*k)*Binomial[2*n,n+2*k],{n,0,10},{k,0,Floor[n/2]}]//TableForm (* Bradley Klee, Feb 26 2018 *)

T(n,k)=(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1)

G.f.: A(x,y) = f/(1-x^2*y*f^4), where f=(1-sqrt(1-4*x))/(2*x) is the Catalan g.f. (A000108).
Row sums equal A088218(n) = C(2*n-1,n).
T(n,0) = A000108(n) (the Catalan numbers).
T(n,1) = A000344(n).
T(n,2) = A001392(n).
Sum_{k=0..floor(n/2)} k*T(n,k) = A000346(n-2).
Eigenvector is defined by: A119244(n) = Sum_{k=0..[n\2]} T(n,k)*A119244(k).
...
T(n,k) = (4*k+1)/(n+2*k+1)*binomial(2*n,n+2*k). Compare with A158483. - Peter Bala, Mar 20 2009
T(n,k) = A039599(n, 2*k). - Johannes W. Meijer, Sep 04 2013
A002894(n) = Sum_{k=0..floor(n/2)} (binomial(2k,k)^2)*(4^(n-2*k))*T(n,k). - Bradley Klee, Feb 26 2018

A026015 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

Original entry on oeis.org

1, 8, 45, 219, 987, 4248, 17748, 72675, 293436, 1172908, 4653935, 18366075, 72186075, 282861360, 1105877880, 4316224860, 16825024134, 65525448960, 255024693434, 992116674142, 3858537980286, 15004402265424, 58343871881400

Offset: 3

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 3..1000

First differences of A001392.

Cf. A000108, A026009.

[Binomial(2*n, n-3) - Binomial(2*n, n-6): n in [3..30]]; // G. C. Greubel, Mar 19 2021

Table[Binomial[2*n, n-3] - Binomial[2*n, n-6], {n, 3, 30}] (* G. C. Greubel, Mar 19 2021 *)

[binomial(2*n, n-3) - binomial(2*n, n-6) for n in (3..30)] # G. C. Greubel, Mar 19 2021

-(n+6)*(n-3)*a(n) +2*(3*n^2+3*n-20)*a(n-1) +(-9*n^2+15*n+20)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*((1-3*x)*(1 -6*x +9*x^2 -3*x^3) -(1-x)*(1 -6*x +9*x^2 -x^3)*sqrt(1-4*x))/(2*x^6).
G.f.: (1-x)*x^3*C(x)^9, where C(x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: exp(2*x)*(BesselI(3, 2*x) - BesselI(6, 2*x)).
a(n) = binomial(2*n, n-3) - binomial(2*n, n-6) = A026009(2*n, n-3).
a(n) = f(n) - f(n-1), where f(n) = Sum_{j=0..n-3} C(n-j-3)*(C(j+7) -6*C(j+6) +10*C(j+5) -4*C(j+4)) and C(n) are the Catalan numbers. (End)
From G. C. Greubel, Mar 22 2021: (Start)
a(n) = C(n+5) -8*C(n+4) +22*C(n+3) -25*C(n+2) +11*C(n+1) -C(n).
a(n) = (9/20)*(binomial(n,3)/binomial(n+6,5))*(3*n^2 +3*n +20)*C(n). (End)

A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 9, 4, 1, 14, 28, 14, 6, 1, 42, 90, 48, 27, 7, 1, 132, 297, 165, 110, 35, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1

Offset: 0

Philippe Deléham, Feb 01 2014

Row sums are A109262(n+1).

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    5,   9,   4,   1;
   14,  28,  14,   6,  1;
   42,  90,  48,  27,  7, 1;
  132, 297, 165, 110, 35, 9, 1;
Production matrix is:
  1...1
  1...2...1
  0...1...1...1
  0...1...1...2...1
  0...0...0...1...1...1
  0...0...0...1...1...2...1
  0...0...0...0...0...1...1...1
  0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...1...1...1
  0...0...0...0...0...0...0...1...1...2...1
  0...0...0...0...0...0...0...0...0...1...1...1
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
Cf. A026674, A026726, A032766, A109262.
Cf. A000045, A039599, A106566.

F:=Factorial;
A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
[A236843(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 13 2022

T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)

T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ Andrew Howroyd, Jan 04 2023

F=factorial
def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 13 2022

G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - G. C. Greubel, Jun 13 2022

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}; ...

cp's OEIS Frontend

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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A119245 Triangle, read by rows, defined by: T(n,k) = (4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1) for n >= 2*k >= 0.

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A026015 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

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A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

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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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A119245 Triangle, read by rows, defined by: T(n,k) = (4k+1)binomial(2n+1, n-2k)/(2n+1) for n >= 2k >= 0.