A234950 -id:A234950 - cp's OEIS frontend

A244887 Third column of triangle in A234950.

2, 20, 135, 770, 4004, 19656, 92820, 426360, 1918620, 8498776, 37182145, 161056350, 691945800, 2952675600, 12527780760, 52895074320, 222399744300, 931689977400, 3890668331550, 16201562020644, 67298796085752, 278927990831600, 1153747598439800, 4763749454427600, 19637233862140440

Offset: 2

N. J. A. Sloane, Jul 12 2014

nonn

Remmel (2014) asks for a formula.

Vincenzo Librandi, Table of n, a(n) for n = 2..500
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.

Cf. A234950.

Table[Sum[Binomial[s, 2] Binomial[n+s, n] (n - s + 1) / (n + 1), {s, 2, n}], {n, 2, 15}] (* Vincenzo Librandi, Apr 06 2018 *)

a(n) = sum(s=2, n, binomial(s, 2)*binomial(n+s, n)*(n-s+1)/(n+1)); \\ Michel Marcus, Apr 06 2018

a(n) = A234950(n, 2).

More terms from Michel Marcus, Apr 06 2018

A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

Original entry on oeis.org

1, 2, -1, 5, -6, 2, 14, -28, 20, -5, 42, -120, 135, -70, 14, 132, -495, 770, -616, 252, -42, 429, -2002, 4004, -4368, 2730, -924, 132, 1430, -8008, 19656, -27300, 23100, -11880, 3432, -429, 4862, -31824, 92820, -157080, 168300, -116688, 51051, -12870, 1430

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) = Sum_{k=0..n} T(n,k)*x^k.
For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992.
Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938.
The positive triangle is |T(n,k)| = binomial(2*n+2, n-k)*binomial(n+k, k)/(n+1). - Paul Barry, May 11 2005

			The triangle N2 = {a(n,k)} begins:
n\k      0       1      2       3       4       5      6       7     8     9
----------------------------------------------------------------------------
0:       1
1:       2      -1
2:       5      -6      2
3:      14     -28     20      -5
4:      42    -120    135     -70      14
5:     132    -495    770    -616     252     -42
6:     429   -2002   4004   -4368    2730    -924    132
7:    1430   -8008  19656  -27300   23100  -11880   3432    -429
8:    4862  -31824  92820 -157080  168300 -116688  51051  -12870  1430
9:   16796 -125970 426360 -852720 1108536 -969969 570570 -217360 48620 -4862
... formatted by _Wolfdieter Lang_, Jan 20 2020
N(2; 2, x)= 5 - 6*x + 2*x^2.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, 2013.
V. E. Hoggatt Jr. and Marjorie Bicknell-Johnson, Numerator Polynomial Coefficient Arrays for Catalan and Related Sequence Convolution Triangles, The Fibonacci Quarterly 15 (1977) 30-34. [On p. 31, in the line n = 1, 14 is missing in S_1^4. - _Wolfdieter Lang_, Jan 20 2020 ]
Joseph T. Iosue, Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov, Page curves and typical entanglement in linear optics, arXiv:2209.06838 [quant-ph], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A000108, A002694, A009766, A033184, A062992, A084938, A089434, A094385.
For an unsigned version see Borel's triangle, A234950.
Sums include: A000012 (row), A000079 (diagonal), A064062 (signed row), A071356 (signed diagonal).

A062991:= func< n,k | (-1)^k*Binomial(2*n+2,n-k)*Binomial(n+k,k)/(n+1) >;
[A062991(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2024

T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A062991(n,k): return (-1)^k*binomial(2*n+2,n-k)*binomial(n+k,k)/(n+1)
flatten([[A062991(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 27 2024

T(n, k) = [x^k] N(2; n, x) with N(2; n, x) = (N(2; n-1, x) - A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) = 1.
T(n, k) = T(n-1, k+1) + (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); T(n, k) = (-1)^k*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0.
O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. - Peter Bala, Jul 15 2012
From G. C. Greubel, Sep 27 2024: (Start)
Sum_{k=0..n} T(n, k) = A000012(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A064062(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000079(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A071356(n). (End)

A062992 Row sums of unsigned triangle A062991.

Original entry on oeis.org

1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n) = N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.

Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.

Cf. A000108, A062991, A064062, A119259.

a062992 = sum . a234950_row  -- Reinhard Zumkeller, Jan 12 2014

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // G. C. Greubel, Sep 27 2024

Table[2*Sum[(-1)^j*Binomial[2*n-2*j,n-j]/(n-j+1)*2^(n-j), {j,0,n}]-(-1)^n,{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2),n)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ Ralf Stephan

def a(n): return hypergeometric([-n, n+1], [-n-1], 2)
[a(n).hypergeometric_simplify() for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).
G.f.: A(x) = (2*c(2*x) - 1)/(1 + x) with c(x) the g.f. of A000108.
a(n) = (1/(n+1)) * Sum_{k = 0..n} binomial(2*n+2, n-k)*binomial(n+k, k). - Paul Barry, May 11 2005
Rewritten: a(n) = (1 - 2*c(n, -2))*(-1)^(n+1), n >= , with c(n, x) := Sum_{k = 0..n} C(k)*x^k and C(k) := A000108(k) (Catalan). - Wolfdieter Lang, Oct 31 2005
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = hypergeometric([-n, n+1], [-n-1], 2). - Peter Luschny, Nov 30 2014
G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - Peter Bala, Jun 08 2023

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 5, 20, 28, 14, 0, 14, 70, 135, 120, 42, 0, 42, 252, 616, 770, 495, 132, 0, 132, 924, 2730, 4368, 4004, 2002, 429, 0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430, 0, 1430, 12870, 51051, 116688, 168300, 157080, 92820, 31824, 4862

Offset: 0

Philippe Deléham, Jun 03 2004, Jun 14 2007

			Triangle begins:
  1;
  0,   1;
  0,   1,    2;
  0,   2,    6,     5;
  0,   5,   20,    28,    14;
  0,  14,   70,   135,   120,    42;
  0,  42,  252,   616,   770,   495,   132;
  0, 132,  924,  2730,  4368,  4004,  2002,  429;
  0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
B. Derrida, E. Domany and D. Mukamel, A exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs.(20), (21), p. 672.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Variant of A062991, unsigned and transposed.
See also A234950 for another version.
Columns: A000007 (k=0), 2*A001700 (k=1).
Diagonals: A002694 (k=n-1), A000108 (k=n).
Row sums: A064062 (generalized Catalan C(2; n)).
Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.
Cf. A064090, A064091, A064092, A064093, A064094.

A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
[A094385(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2024

T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 26 2024

T is given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.
Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.
From G. C. Greubel, Sep 26 2024: (Start)
T(n, 1) = A000108(n-1), n >= 1.
T(n, n-1) = A002694(n), n >= 1.
T(n, n) = A000108(n). (End)

New name using a formula of the author by Peter Luschny, Sep 26 2024

A277930 Array of coefficients a(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = ((2k+1)x+sqrt(1+4k(k+1)*x^2))/(1-x^2), k>=0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 3, 1, 1, 9, 25, 5, -3, 1, 1, 11, 41, 7, -59, 3, 1, 1, 13, 61, 9, -263, 5, 29, 1, 1, 15, 85, 11, -759, 7, 805, 3, 1, 1, 17, 113, 13, -1739, 9, 6649, 5, -131, 1, 1, 19, 145, 15, -3443, 11, 31241, 7, -12155, 3, 1, 1, 21, 181, 17, -6159, 13, 106261, 9, -200711, 5, 765, 1

Offset: 0

Werner Schulte, Nov 04 2016

The A(k,x) satisfy A(k,x)^2 = 1+(4*k+2)*x*A(k,x)+x^2*A(k,x)^2 for k>=0.

The terms of odd-numbered columns a(k,2*n+1) are simple with (2*k+1)*x/(1-x^2), analogous the even-numbered columns a(k,2*n) with the o.g.f. of A000108.

			The terms define the array a(k,n) for k >= 0 and n >= 0, i.e.,
k\n  0   1    2   3       4   5        6   7           8   9         10  11  ...
0:   1   1    1   1       1   1        1   1           1   1          1   1  ...
1:   1   3    5   3      -3   3       29   3        -131   3        765   3  ...
2:   1   5   13   5     -59   5      805   5      -12155   5     205573   5  ...
3:   1   7   25   7    -263   7     6649   7     -200711   7    6766585   7  ...
4:   1   9   41   9    -759   9    31241   9    -1568759   9   88031241   9  ...
5:   1  11   61  11   -1739  11   106261  11    -7993739  11  672406261  11  ...
6:   1  13   85  13   -3443  13   292909  13   -30824051  13  ...
7:   1  15  113  15   -6159  15   696305  15   -97648655  15  ...
8:   1  17  145  17  -10223  17  1482769  17  -267255791  17  ...
9:   1  19  181  19  -16019  19  2899981  19  ...
10:  1  21  221  21  -23979  21  5300021  21  ...
etc.
The formal power series corresponding to row 2 is A(2,x) = 1+5*x+13*x^2+5*x^3 ..
The terms define the triangle T(k,n) = a(k-n,n) for 0 <= n <=k, i.e.,
k\n  0  1   2  3   4  5  ...
0:   1
1:   1  1
2:   1  3   1
3:   1  5   5  1
4:   1  7  13  3   1
5:   1  9  25  5  -3  1
etc.

Cf. A000108, A234950.

A[k_, n_]:=If[n==0, 1, If[EvenQ[n], 1 - 2 Sum[CatalanNumber[i] (-k(k + 1))^(i + 1), {i, 0, (n - 2)/2}], 2k + 1]]; Table[A[n - k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Indranil Ghosh, Aug 03 2017 *)

from sympy import catalan
def A(k, n): return 1 if n==0 else 1 - 2*sum([catalan(i)*(-k*(k + 1))**(i + 1) for i in range(n//2)]) if n%2==0 else 2*k + 1
for n in range(13): print([A(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Aug 03 2017

a(k,0) = 1 and a(k,2*n+2) = 1-2*(Sum_{i=0..n} A000108(i)*(-k*(k+1))^(i+1)) and a(k,2*n+1) = 2*k+1 for k >= 0 and n >= 0.
A(k,x) = (1+(2*k+1)*x+2*k*(k+1)*x^2*C(-k*(k+1)*x^2))/(1-x^2) for k >= 0, where C is the o.g.f. of A000108.
A(k,x)*A(k,-x) = 1/(1-x^2) for k >= 0.
Conjecture: a(k,2*n+2) = 1+2*k+2*(-k)^(n+2)*(Sum_{i=0..n} A234950(n,i)*k^i) for k>=0 and n>=0. - Werner Schulte, Aug 03 2017

cp's OEIS Frontend

A244887 Third column of triangle in A234950.

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A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x).

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A062992 Row sums of unsigned triangle A062991.

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A094385 Triangle read by rows: T(n, k) = binomial(2*n, k-1)*binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

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A277930 Array of coefficients a(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = ((2*k+1)*x+sqrt(1+4*k*(k+1)*x^2))/(1-x^2), k>=0.

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A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

A277930 Array of coefficients a(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = ((2k+1)x+sqrt(1+4k(k+1)*x^2))/(1-x^2), k>=0.