A062746 -id:A062746 - cp's OEIS frontend

A062747 Row sums of (unsigned) staircase array A062746.

1, 7, 89, 1447, 26713, 532391, 11165785, 242851751, 5427716185, 123901026215, 2876525797465, 67710590623655, 1612262780199001, 38764533106581415, 939825790848884825, 22950405085586497447

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n)=N(3; k, x=-1), with the polynomials N(3; k, x) from the staircase array A062746.
a(n) = 2*( Sum_{j = 0..n} (-1)^j*C(3; n-j)*4^(n-j) ) - (-1)^n with C(3; n) := A001764(n) = A062993(n+1, 1) (a Pfaff-Fuss or 3-Raney sequence).
G.f.: (2*c(3; 4*x)-1)/(1+x) with c(3; x)= RootOf(x*A001764%20%5Bformula%20for%20a(n)%20and%20g.f.%20corrected%20by%20_Peter%20Bala">Z^3-_Z +1), the g.f. of A001764 [formula for a(n) and g.f. corrected by _Peter Bala, Mar 26 2020].
Conjectural recurrence: n*(2*n+1)*a(n) = (4*n-3)*(13*n-4)*a(n-1) + 6*(3*n-1)*(3*n-2)*a(n-2) with a(0) = 1, a(1) = 7. - Peter Bala, Mar 25 2020

A062748 Fourth column (r=3) of FS(3) staircase array A062745.

Original entry on oeis.org

3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, 454, 559, 679, 815, 968, 1139, 1329, 1539, 1770, 2023, 2299, 2599, 2924, 3275, 3653, 4059, 4494, 4959, 5455, 5983, 6544, 7139, 7769, 8435, 9138, 9879, 10659, 11479, 12340, 13243, 14189, 15179, 16214, 17295, 18423

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 3}_{2}, n >= 0.
If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=6, a(n-6) = coeff(charpoly(A,x), x^(n-2)). - Milan Janjic, Jan 26 2010
For n>=4, a(n-4) is the number of permutations of 1,2,...,n, such that n-3 is the only up-point, or, the same, a(n-4) is up-down coefficient {n,4} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014

			G.f. = 3 + 9*x + 19*x^2 + 34*x^3 + 55*x^4 + 83*x^5 + 119*x^6 + 164*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Guillaume Aupy and Julien Herrmann. Periodicity in optimal hierarchical checkpointing schemes for adjoint computations. Optimization Methods and Software, Volume 32, 2017 - Issue 3. Preprint
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A column of triangle A014473.

Cf. A000124, A000292, A050407, A060351, A062745.

[Binomial(n+4,3) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024

seq(((n^3-n)/6)-1,n=3..40); # Zerinvary Lajos, May 05 2007

LinearRecurrence[{4,-6,4,-1},{3,9,19,34},40] (* Harvey P. Dale, Jan 13 2019 *)
Binomial[4+Range[0,50], 3] -1 (* G. C. Greubel, Apr 22 2024 *)

{a(n) = binomial(n+4, 3) - 1}; /* Michael Somos, Jan 28 2018 */

[binomial(n+4,3) - 1 for n in range(51)] # G. C. Greubel, Apr 22 2024

a(n) = A062745(n+2, 3) = binomial(n+4, 3) - 1 = (n+1)*(n^2 + 8*n + 18)/3!.
G.f.: N(3;1, x)/(1-x)^4 with N(3;1, x) = 3 - 3*x + x^2, polynomial of the second row of A062746.
a(n-3) = ((n^3 - n)/6) - 1, n >= 3. - Zerinvary Lajos, May 05 2007
a(n) = A000292(n+2) - 1. - Zerinvary Lajos, May 05 2007
a(n) = Sum_{i=2..n} i*(i+1)/2. - Artur Jasinski, Mar 14 2008
a(n) = -A050407(-1-n) for all n in Z. - Michael Somos, Jan 28 2018
a(n) = A000292(n+3) - A000124(n+3). - Torlach Rush, Aug 03 2018
E.g.f.: (1/6)*(18 + 36*x + 12*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 22 2024

A062745 Generalized Catalan array FS(3; n,r).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 6, 9, 12, 12, 12, 1, 4, 10, 19, 31, 43, 55, 55, 55, 1, 5, 15, 34, 65, 108, 163, 218, 273, 273, 273, 1, 6, 21, 55, 120, 228, 391, 609, 882, 1155, 1428, 1428, 1428, 1, 7, 28, 83, 203, 431, 822, 1431, 2313, 3468, 4896, 6324, 7752, 7752

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this array appears in Table 2, p. 143 and is called {n over r}_{m-1}, with m=3.
The step width sequence of this staircase array is [1,2,2,2,....], i.e., the degree of the row polynomials is [0,2,4,6,...] = A005843.
The columns r=0..5 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A062748, A005718, A062749.
Number of lattice paths from (0,0) to (r,n) using steps h=(1,0), v=(0,1) and staying on or above the line y = x/2. Example: a(3,2)=6 because from (0,0) to (2,3) we have the following valid paths: vvvhh, vvhvh, vvhhv, vhvvh, vhvhvh and vhvvh (see the Niederhausen reference). - Emeric Deutsch, Jun 24 2005

			Array begins:
  {1};
  {1,1,1};
  {1,2,3,3,3};
  {1,3,6,9,12,12,12};
  ...;
N(3; 1,x) = 3-3*x+x^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Wolfdieter Lang, First 10 rows.
Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table 2).
Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.

Cf. A009766, A280759 (rows reversed).
Cf. A062746, A062747, A062748, A062749.
Cf. A005843, A000012, A000027, A000217, A005718.

a:=proc(n,r) if r<=2*n then binomial(n+r,r)-(-1)^(r-1)*sum(binomial(3*i,i)*binomial(i-n-1,r-1-2*i)/(2*i+1),i=0..floor((r-1)/2)) else 0 fi end: for n from 0 to 8 do seq(a(n,r),r=0..2*n) od; # yields sequence in triangular form # Emeric Deutsch, Jun 24 2005

a[0, 0] = 1; a[, -1] = 0; a[n, r_] /; r > 2*n = 0; a[n_, r_] := a[n, r] = a[n, r-1] + a[n-1, r]; Table[a[n, r], {n, 0, 7}, {r, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

a(0,0)=1, a(n,-1)=0, n >= 1; a(n,r) = a(n, r-1) + a(n-1, r) if r <= 2n, 0 otherwise.
G.f. for column r = 2*k+j, k >= 0, j=1, 2: (x^(k+1))*N(3; k, x)/ (1-x)^(2*k+1+j), with row polynomials N(3; k, x) of array A062746; for column r=0: 1/(1-x).
a(n,r) = binomial(n+r, r) - (-1)^(r-1)*Sum_{i=0..floor((r-1)/2)} binomial(3i, i)*binomial(i-n-1, r-1-2i)/(2i+1), 0 <= r <= 2n (see the Niederhausen reference, eq. (17)). - Emeric Deutsch, Jun 24 2005

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

Original entry on oeis.org

1, 5, -10, 10, -5, 1, 35, -170, 415, -629, 630, -420, 180, -45, 5, 285, -2315, 9381, -24395, 44625, -59880, 60015, -45040, 25025, -10010, 2730, -455, 35, 2530, -29379, 169405, -633675, 1703700, -3467145, 5497640, -6903325

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=4*k+j, k >= 0, j=1,2,3,4, of the staircase array A062985(n,r) is N(5; k,x)*(x^(k+1))/(1-x)^(4*k+1+j) with N(5; k,x) := sum(a(k,p)*x^p,p=0..4*k).
The m=0 column gives A002294(k+1). The row sums give A000012 (powers of 1) and (unsigned) A062987.
The sequence of step width of this staircase array is [1,4,4,4,...], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.

			{1}; {5,-10,10,-5,1}; {35,-170,415,-629,630,-420,180,-45,5}; ...; N(5; 1,x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x.

A062991, A062746, A062751.

a(k, p) := [x^p]N(5; k, x) with N(5; k, x)=(N(5; k-1, x)- A002294(k)*(1-x)^(4*k+1))/x, N(5; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=0, .., (4*n-5); a(n, k)= ((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=(4*n-4), ..., 4*n; else 0.

A062749 Sixth column (r=5) of FS(3) staircase array A062745.

Original entry on oeis.org

12, 43, 108, 228, 431, 753, 1239, 1944, 2934, 4287, 6094, 8460, 11505, 15365, 20193, 26160, 33456, 42291, 52896, 65524, 80451, 97977, 118427, 142152, 169530, 200967, 236898, 277788, 324133, 376461

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+3) over 5}_{2}, n >= 0.

Colin Barker, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

seq(coeff(series((3*x^4-15*x^3+30*x^2-29*x+12)/(1-x)^6,x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 30 2018

Vec((12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Oct 30 2018

a(n) = A062745(n+3, 5)= -3+binomial(n+4, 3)*(n^2+16*n+75)/20 = (n+1)*(n^4+24*n^3+221*n^2+894*n+1440)/5!.
G.f.: N(3;2, x)/(1-x)^6 with N(3;2, x)= 12-29*x+30*x^2-15*x^3+3*x^4, polynomial of the third row of A062746.
From Colin Barker, Oct 30 2018: (Start)
G.f.: (12 - 29*x + 30*x^2 - 15*x^3 + 3*x^4) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
(End)

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A062747 Row sums of (unsigned) staircase array A062746.

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A062748 Fourth column (r=3) of FS(3) staircase array A062745.

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A062745 Generalized Catalan array FS(3; n,r).

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A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

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A062749 Sixth column (r=5) of FS(3) staircase array A062745.

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