A225010 -id:A225010 - cp's OEIS frontend

A059259 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...

1, 1, 0, 1, 1, 1, 1, 2, 2, 0, 1, 3, 4, 2, 1, 1, 4, 7, 6, 3, 0, 1, 5, 11, 13, 9, 3, 1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, 10, 46, 128, 239, 314, 296, 200

Offset: 0

N. J. A. Sloane, Jan 23 2001

This sequence provides the general solution to the recurrence a(n) = a(n-1) + k*(k+1)*a(n-2), a(0)=a(1)=1. The solution is (1, 1, k^2 + k + 1, 2*k^2 + 2*k + 1, ...) whose coefficients can be read from the rows of the triangle. The row sums of the triangle are given by the case k=1. These are the Jacobsthal numbers, A001045. Viewed as a square array, its first row is (1,0,1,0,1,...) with e.g.f. cosh(x), g.f. 1/(1-x^2) and subsequent rows are successive partial sums given by 1/((1-x)^n * (1-x^2)). - Paul Barry, Mar 17 2003
Conjecture: every second column of this triangle is identical to a column in the square array A071921. For example, column 4 of A059259 (1, 3, 9, 22, 46, ...) appears to be the same as column 3 of A071921; column 6 of A059259 (1, 4, 16, 50, 130, 296, ...) appears to be the same as column 4 of A071921; and in general column 2k of A059259 appears to be the same as column k+1 of A071921. Furthermore, since A225010 is a transposition of A071921 (ignoring the latter's top row and two leftmost columns), there appears to be a correspondence between column 2k of A059259 and row k of A225010. - Mathew Englander, May 17 2014
T(n,k) is the number of n-tilings of a (one-dimensional) board that use k (1,1)-fence tiles and n-k squares. A (1,1)-fence is a tile composed of two pieces of width 1 separated by a gap of width 1. - Michael A. Allen, Jun 25 2020
See the Edwards-Allen 2020 paper, page 14, for proof of Englander's conjecture. - Michael De Vlieger, Dec 10 2020

			Triangle begins:
  1;
  1,  0;
  1,  1,  1;
  1,  2,  2,   0;
  1,  3,  4,   2,   1;
  1,  4,  7,   6,   3,   0;
  1,  5, 11,  13,   9,   3,   1;
  1,  6, 16,  24,  22,  12,   4,   0;
  1,  7, 22,  40,  46,  34,  16,   4,  1;
  1,  8, 29,  62,  86,  80,  50,  20,  5,  0;
  1,  9, 37,  91, 148, 166, 130,  70, 25,  5, 1;
  1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0;
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, Frogs, hats and common subsequences, arXiv:2404.07285 [math.CO], 2024. See p. 28.
Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.

See A059260 for an explicit formula.
Diagonals of this triangle are given by A006498.
Similar to the triangles A035317, A080242, A108561, A112555.
Cf. A123521, A157897.

read transforms; 1/(1-x-x*y-y^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);

T[n_, 0]:= 1; T[n_, n_]:= (1+(-1)^n)/2; T[n_, k_]:= T[n, k] = T[n-1, k] + T[n-1, k-1]; Table[T[n, k], {n, 0, 10} , {k, 0, n}]//Flatten (* G. C. Greubel, Jan 03 2017 *)

{T(n,k) = if(k==0, 1, if(k==n, (1+(-1)^n)/2, T(n-1,k) +T(n-1,k-1)) )};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 29 2019

def A059259_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return (-1)^n
        if k==0: return 0
        return prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n)]
for n in (1..12): print(A059259_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 1/(1 - x - x*y - y^2).
As a square array read by antidiagonals, this is T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(i+k, k). - Paul Barry, Jul 01 2003
T(2*n,n) = A026641(n). - Philippe Deléham, Mar 08 2007
T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = T(2,2)=1, T(1,1)=0, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 24 2013
T(n,0) = 1, T(n,n) = (1+(-1)^n)/2, and T(n,k) = T(n-1,k) + T(n-1,k-1) for 0 < k < n. - Mathew Englander, May 24 2014
From Michael A. Allen, Jun 25 2020: (Start)
T(n,k) + T(n-1,k-1) = binomial(n,k) if n >= k > 0.
T(2*n-1,2*n-2) = T(2*n,2*n-1) = n, T(2*n,2*n-2) = n^2, T(2*n+1,2*n-1) = n*(n+1) for n > 0.
T(n,2) = binomial(n-2,2) + n - 1 for n > 1 and T(n,3) = binomial(n-3,3) + 2*binomial(n-2,2) for n > 2.
T(2*n-k,k) = A123521(n,k). (End)

A071920 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=0 for all m>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 9, 7, 1, 0, 0, 5, 16, 22, 11, 1, 0, 0, 6, 25, 50, 46, 16, 1, 0, 0, 7, 36, 95, 130, 86, 22, 1, 0, 0, 8, 49, 161, 295, 296, 148, 29, 1, 0, 0, 9, 64, 252, 581, 791, 610, 239, 37, 1, 0

Offset: 0

Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002

If one uses a definition of unimodality that involves existential quantifiers on the domain of a function then a(0,m)=0 a priori.

			Square array a(n,m) begins:
  0,    0,    0,    0,    0,    0,     0,     0,      0, ...
  0,    1,    2,    3,    4,    5,     6,     7,      8, ...
  0,    1,    4,    9,   16,   25,    36,    49,     64, ...
  0,    1,    7,   22,   50,   95,   161,   252,    372, ...
  0,    1,   11,   46,  130,  295,   581,  1036,   1716, ...
  0,    1,   16,   86,  296,  791,  1792,  3612,   6672, ...
  0,    1,   22,  148,  610, 1897,  4900, 11088,  22716, ...
  0,    1,   29,  239, 1163, 4166, 12174, 30738,  69498, ...
  0,    1,   37,  367, 2083, 8518, 27966, 78354, 194634, ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A071921, A225010.

Main diagonal is A088536.

a:= (n, m)-> `if`(n=0, 0, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
seq(seq(a(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 21 2013

a[n_, m_] := Sum[Binomial[n+2*k-1, 2*k], {k, 0, m-1}]; a[0, ] = 0; Table[a[n-m, m], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* _Jean-François Alcover, Feb 25 2015 *)

a(n,m) = Sum_{k=0..m-1} binomial(n+2k-1, 2k) if n>0.

A071921 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 7, 1, 0, 1, 5, 16, 22, 11, 1, 0, 1, 6, 25, 50, 46, 16, 1, 0, 1, 7, 36, 95, 130, 86, 22, 1, 0, 1, 8, 49, 161, 295, 296, 148, 29, 1, 0, 1, 9, 64, 252, 581, 791, 610, 239, 37, 1, 0

Offset: 0

Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002

If one uses a definition of unimodality that involves universal quantifiers on the domain of a function then a(0,m)=1 a priori.

			Square array a(n,m) begins:
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  0, 1,  2,   3,    4,    5,     6,     7,      8, ...
  0, 1,  4,   9,   16,   25,    36,    49,     64, ...
  0, 1,  7,  22,   50,   95,   161,   252,    372, ...
  0, 1, 11,  46,  130,  295,   581,  1036,   1716, ...
  0, 1, 16,  86,  296,  791,  1792,  3612,   6672, ...
  0, 1, 22, 148,  610, 1897,  4900, 11088,  22716, ...
  0, 1, 29, 239, 1163, 4166, 12174, 30738,  69498, ...
  0, 1, 37, 367, 2083, 8518, 27966, 78354, 194634, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kenneth Edwards and Michael A. Allen, New Combinatorial Interpretations of the Fibonacci Numbers Squared, Golden Rectangle Numbers, and Jacobsthal Numbers Using Two Types of Tile, arXiv:2009.04649 [math.CO], 2020.

Cf. A071920, A225010.

Main diagonal gives A088536 (for n>=1).

a:= (n, m)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..m-1)):
seq(seq(a(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Sep 22 2013

a[0, 0] = 1; a[n_, m_] := Sum[Binomial[2k+n-1, 2k], {k, 0, m-1}]; Table[a[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 11 2015 *)

a(n,m) = 1 if n=0, m>=0, a(n,m) = Sum_{k=0..m-1} C(2k+n-1,2k) otherwise.

A239473 Triangle read by rows: signed version of A059260: coefficients for expansion of partial sums of sequences a(n,x) in terms of their binomial transforms (1+a(.,x))^n ; Laguerre polynomial expansion of the truncated exponential.

Original entry on oeis.org

1, 0, 1, 1, -1, 1, 0, 2, -2, 1, 1, -2, 4, -3, 1, 0, 3, -6, 7, -4, 1, 1, -3, 9, -13, 11, -5, 1, 0, 4, -12, 22, -24, 16, -6, 1, 1, -4, 16, -34, 46, -40, 22, -7, 1, 0, 5, -20, 50, -80, 86, -62, 29, -8, 1, 1, -5, 25, -70, 130, -166, 148, -91, 37, -9, 1, 0, 6, -30, 95, -200, 296, -314, 239, -128, 46, -10, 1

Offset: 0

Tom Copeland, Mar 19 2014

With T the lower triangular array above and the Laguerre polynomials L(k,x) = Sum_{j=0..k} (-1)^j binomial(k, j) x^j/j!, the following identities hold:
(A) Sum_{k=0..n} (-1)^k L(k,x) = Sum_{k=0..n} T(n,k) x^k/k!;
(B) Sum_{k=0..n} x^k/k! = Sum_{k=0..n} T(n,k) L(k,-x);
(C) Sum_{k=0..n} x^k = Sum_{k=0..n} T(n,k) (1+x)^k = (1-x^(n+1))/(1-x).
More generally, for polynomial sequences,
(D) Sum_{k=0..n} P(k,x) = Sum_{k=0..n} T(n,k) (1+P(.,x))^k,
where, e.g., for an Appell sequence, such as the Bernoulli polynomials, umbrally, (1+ Ber(.,x))^k = Ber(k,x+1).
Identity B follows from A through umbral substitution of j!L(j,-x) for x^j in A. Identity C, related to the cyclotomic polynomials for prime index, follows from B through the Laplace transform.
Integrating C gives Sum_{k=0..n} T(n,k) (2^(k+1)-1)/(k+1) = H(n+1), the harmonic numbers.
Identity A >= 0 for x >= 0 (see MathOverflow link for evaluation in terms of Hermite polynomials).
From identity C, W(m,n) = (-1)^n Sum_{k=0..n} T(n,k) (2-m)^k = number of walks of length n+1 between any two distinct vertices of the complete graph K_m for m > 2.
Equals A112468 with the first column of ones removed. - Georg Fischer, Jul 26 2023

			Triangle begins:
   1
   0    1
   1   -1    1
   0    2   -2    1
   1   -2    4   -3    1
   0    3   -6    7   -4    1
   1   -3    9  -13   11   -5    1
   0    4  -12   22  -24   16   -6    1
   1   -4   16  -34   46  -40   22   -7    1
   0    5  -20   50  -80   86  -62   29   -8    1
   1   -5   25  -70  130 -166  148  -91   37   -9    1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J. Adams, On the groups J(x)-II, Topology, Vol. 3, p. 137-171, Pergamon Press, (1965).
MathOverflow, Cyclotomic Polynomials in Combinatorics
Mathoverflow, Inequality for Laguerre polynomials

For column 2: A001057, A004526, A008619, A140106.
Column 3: A002620, A087811.
Column 4: A002623, A173196.
Column 5: A001752.
Column 6: A001753.
Cf. Bottomley's cross-references in A059260.
Embedded in alternating antidiagonals of T are the reversals of arrays A071921 (A225010) and A210220.
Cf. A049310, A112468, A341091.

[[(&+[(-1)^(j+k)*Binomial(j,k): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

A239473 := proc(n,k)
    add(binomial(j,k)*(-1)^(j+k),j=k..n) ;
end proc; # R. J. Mathar, Jul 21 2016

Table[Sum[(-1)^(j+k)*Binomial[j,k], {j,0,n}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2018 *)

for(n=0,10, for(k=0,n, print1(sum(j=0,n, (-1)^(j+k)*binomial(j, k)), ", "))) \\ G. C. Greubel, Feb 06 2018

Trow = lambda n: sum((x-1)^j for j in (0..n)).list()
for n in (0..10): print(Trow(n)) # Peter Luschny, Jul 09 2019

T(n, k) = Sum_{j=0..n} (-1)^(j+k) * binomial(j, k).
E.g.f: (exp(t) - (x-1)*exp((x-1)*t))/(2-x).
O.g.f. (n-th row): (1-(x-1)^(n+1))/(2-x).
Associated operator identities:
With D=d/dx, :xD:^n=x^n*D^n, and :Dx:^n=D^n*x^n, then bin(xD,n)= binomial(xD,n)=:xD:^n/n! and L(n,-:xD:)=:Dx:^n/n!=bin(xD+n,n)=(-1)^n bin(-xD-1,n),
A-o) Sum_{k=0..n} (-1)^k L(k,-:xD:) = Sum_{k=0..n} :-Dx:^k/k!
     = Sum_{k=0..n} T(n,k) :-xD:^k/k! = Sum_{k=0..n} (-1)^k T(n,k)bin(xD,k)
B-o) Sum_{k=0..n} :xD:^k/k! = Sum_{k=0..n}, T(n,k) L(k,-:xD:)
     = Sum_{k=0..n} T(n,k) :Dx:^k/k! = Sum_{k=0..n}, bin(xD,k).
Associated binomial identities:
A-b) Sum_{k=0..n} (-1)^k bin(s+k,k) = Sum_{k=0..n} (-1)^k T(n,k) bin(s,k)
     = Sum_{k=0..n} bin(-s-1,k) = Sum{k=0..n} T(n,k) bin(-s-1+k,k)
B-b) Sum_{k=0..n} bin(s,k) = Sum_{k=0..n} T(n,k) bin(s+k,k)
     = Sum_{k=0..n} (-1)^k bin(-s-1+k,k)
     = Sum_{k=0..n} (-1)^k T(n,k) bin(-s-1,k).
In particular, from B-b with s=n, Sum_{k=0..n} T(n,k) bin(n+k,k) = 2^n. From B-b with s=0, row sums are all 1.
From identity C with x=-2, the unsigned row sums are the Jacobsthal sequence, i.e., Sum_{k=0..n} T(n,k) (1+(-2))^k = (-1)^n A001045(n+1); for x=2, the Mersenne numbers A000225; for x=-3, A014983 or signed A015518; for x=3, A003462; for x=-4, A014985 or signed A015521; for x=4, A002450; for x=-5, A014986 or signed A015531; and for x=5, A003463; for x=-6, A014987 or signed A015540; and for x=6, A003464.
With -s-1 = m = 0,1,2,..., B-b gives finite differences (recursions):
Sum_{k=0..n} (-1)^k T(n,k) bin(m,k) = Sum_{k=0..n} (-1)^k bin(m+k,k) = T(n+m,m), i.e., finite differences of the columns of T generate shifted columns of T. The columns of T are signed, shifted versions of sequences listed in the cross-references. Since the finite difference is an involution, T(n,k) = Sum_{j=0..k} (-1)^j T(n+j,j) bin(k,j)}. Gauss-Newton interpolation can be applied to give a generalized T(n,s) for s noninteger.
From identity C, S(n,m) = Sum_{k=0..n} T(n,k) bin(k,m) = 1 for m < n+1 and 0 otherwise, i.e., S = T*P, where S = A000012, as a lower triangular matrix and P = Pascal = A007318, so T = S*P^(-1), where P^(-1) = A130595, the signed Pascal array (see A132440), the inverse of P, and T^(-1) = P*S^(-1) = P*A167374 = A156644.
U(n,cos(x)) = e^(-n*i*x)*Sum_{k=0..n} T(n,k)*(1+e^(2*i*x))^k = sin((n+1)x)/sin(x), where U is the Chebyschev polynomial of the second kind A053117 and i^2 = -1. - Tom Copeland, Oct 18 2014
From Tom Copeland, Dec 26 2015: (Start)
With a(n,x) = e^(nx), the partial sums are 1+e^x+...+e^(nx) = Sum_{k=0..n} T(n,k) (1+e^x)^k = [ x / (e^x-1) ] [ e^((n+1)x) -1 ] / x = [ (x / (e^x-1)) e^((n+1)x) -  (x / (e^x-1)) ] / x =  Sum_{k>=0} [ (Ber(k+1,n+1) - Ber(k+1,0)) / (k+1) ] * x^k/k!, where Ber(n,x) are the Bernoulli polynomials (cf. Adams p. 140). Evaluating (d/dx)^m at x=0 of these expressions gives relations among the partial sums of the m-th powers of the integers, their binomial transforms, and the Bernoulli polynomials.
With a(n,x) = (-1)^n e^(nx), the partial sums are 1-e^x+...+(-1)^n e^(nx) = Sum_{k=0..n} T(n,k) (1-e^x)^k = [ (-1)^n e^((n+1)x) + 1 ] / (e^x+1) = [ (-1)^n (2 / (e^x+1)) e^((n+1)x) + (2 / (e^x+1)) ] / 2 = (1/2) Sum_{k>=0} [ (-1)^n Eul(k,n+1) + Eul(k,0) ] * x^k/k!, where Eul(n,x) are the Euler polynomials. Evaluating (d/dx)^m at x=0 of these expressions gives relations among the partial sums of signed m-th powers of the integers; their binomial transforms, related to the Stirling numbers of the second kind and face numbers of the permutahedra; and the Euler polynomials. (End)
As in A059260, a generator in terms of bivariate polynomials with the coefficients of this entry is given by (1/(1-y))*1/(1 + (y/(1-y))*x - (1/(1-y))*x^2) = 1 + y + (x^2 - x*y + y^2) + (2*x^2*y - 2*x*y^2 + y^3) + (x^4 - 2*x^3*y + 4*x^2*y^2 - 3*x*y^3 + y^4) + ... . This is of the form -h2 * 1 / (1 + h1*x + h2*x^2), related to the bivariate generator of A049310 with h1 = y/(1-y) and h2 = -1/(1-y) = -(1+h1). - Tom Copeland, Feb 16 2016
From Tom Copeland, Sep 05 2016: (Start)
Letting P(k,x) = x in D gives Sum_{k=0..n} T(n,k)*Sum_{j=0..k} binomial(k,j) = Sum_{k=0..n} T(n,k) 2^k = n + 1.
The quantum integers [n+1]q = (q^(n+1) - q^(-n-1)) / (q - q^(-1)) = q^(-n)*(1 - q^(2*(n+1))) / (1 - q^2) = q^(-n)*Sum{k=0..n} q^(2k) = q^(-n)*Sum_{k=0..n} T(n,k)*(1 + q^2)^k. (End)
T(n, k) = [x^k] Sum_{j=0..n} (x-1)^j. - Peter Luschny, Jul 09 2019
a(n) = -n + Sum_{k=0..n} A341091(k). - Thomas Scheuerle, Jun 17 2022

Inverse array added by Tom Copeland, Mar 26 2014

Formula re Euler polynomials corrected by Tom Copeland, Mar 08 2024

A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

1, 2, 9, 50, 295, 1792, 11088, 69498, 439791, 2803658, 17978389, 115837592, 749321716, 4863369656, 31655226108, 206549749930, 1350638103791, 8848643946550, 58069093513635, 381650672631330, 2511733593767295, 16550500379912640, 109176697072162080

Offset: 0

R. H. Hardin, Apr 23 2013

nonn

Diagonal of A225010.

Number of unimodal maps [1..n]->[1..n+1], see example. - Joerg Arndt, May 10 2013

			Some solutions for n=3
..0..1..1....0..1..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..0
..1..1..1....0..1..0....1..1..1....0..0..0....0..0..0....0..1..0....0..0..1
..1..1..1....0..1..1....1..1..1....0..0..1....0..1..0....1..1..1....0..1..1
From _Joerg Arndt_, May 10 2013: (Start)
The a(2) = 9 unimodal maps [1,2]->[1,2,3] are
01:  [ 1 1 ]
02:  [ 1 2 ]
03:  [ 1 3 ]
04:  [ 2 1 ]
05:  [ 2 2 ]
06:  [ 2 3 ]
07:  [ 3 1 ]
08:  [ 3 2 ]
09:  [ 3 3 ]
(End)

G. C. Greubel and R. H. Hardin, Table of n, a(n) for n = 0..1000 (terms 1..51 from R. H. Hardin)

Cf. A088536 (unimodal maps [1..n]->[1..n]).

Cf. A183160, A371798.

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 0, n}]; Array[a, 30] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

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

From Vaclav Kotesovec, May 22 2013: (Start)
Empirical: 4*n*(2*n-1)*(5*n-7)*a(n) = 2*(145*n^3 - 343*n^2 + 235*n - 48)*a(n-1) - 3*(3*n-4)*(3*n-2)*(5*n-2)*a(n-2).
a(n) ~ 3^(3*n+3/2)/(5*2^(2*n+1)*sqrt(Pi*n)). (End)
a(n) = A261668(n)+1.
a(n) = Sum_{d=0..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in (1+x)^(-n-1)/(1-x). - Max Alekseyev, Sep 14 2015
It appears that a(n) = Sum_{k = 0..2*n} (-1)^k*binomial(n+k,k). - Peter Bala, Oct 08 2021
From Seiichi Manyama, Apr 06 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1+x^2) * (1-x)^(2*n)). (End)

a(0)=1 prepended by Alois P. Heinz, Feb 04 2017

A225007 Number of n X 5 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

1, 16, 86, 296, 791, 1792, 3612, 6672, 11517, 18832, 29458, 44408, 64883, 92288, 128248, 174624, 233529, 307344, 398734, 510664, 646415, 809600, 1004180, 1234480, 1505205, 1821456, 2188746, 2613016, 3100651, 3658496, 4293872, 5014592, 5828977

Offset: 0

R. H. Hardin, Apr 23 2013

			Some solutions for n=3:
  0 0 0 0 0    0 0 1 0 0    0 0 1 1 1    0 0 1 1 0
  0 0 0 1 0    0 1 1 0 0    0 0 1 1 1    0 0 1 1 1
  0 0 1 1 0    0 1 1 1 0    0 0 1 1 1    0 1 1 1 1
6*a(7) = 40032 = 1*6 + 6*15 + 15*28 + 28*45 + 45*66 + 66*91 + 91*120 + 120*153. - _Bruno Berselli_, Feb 05 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 1..210 from R. H. Hardin)

Column 5 of A225010.

a(n) = (2/15)*n^5 + (7/6)*n^4 + (23/6)*n^3 + (35/6)*n^2 + (121/30)*n + 1.
6*a(n) = Sum_{i=1..n+1} A000384(i)*A000384(i+1). - Bruno Berselli, Feb 05 2014
From Colin Barker, Mar 16 2018: (Start)
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 >= 6.
G.f.: (1 + 10*x + 5*x^2) / (1 - x)^6. (End)

a(0)=1 prepended by Andrew Howroyd, Feb 11 2024

A225008 Number of n X 6 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

22, 148, 610, 1897, 4900, 11088, 22716, 43065, 76714, 129844, 210574, 329329, 499240, 736576, 1061208, 1497105, 2072862, 2822260, 3784858, 5006617, 6540556, 8447440, 10796500, 13666185, 17144946, 21332052, 26338438, 32287585, 39316432

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

			Some solutions for n=3:
..0..1..0..0..0..0....0..0..1..0..0..0....0..0..1..1..1..0....0..0..1..1..0..0
..0..1..0..0..0..0....0..1..1..1..1..0....0..0..1..1..1..0....0..1..1..1..1..0
..0..1..1..1..0..0....1..1..1..1..1..0....0..1..1..1..1..1....0..1..1..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 6 of A225010.

Empirical: a(n) = (2/45)*n^6 + (8/15)*n^5 + (91/36)*n^4 + 6*n^3 + (1337/180)*n^2 + (67/15)*n + 1.
Conjectures from Colin Barker, Sep 05 2018: (Start)
G.f.: x*(22 - 6*x + 36*x^2 - 35*x^3 + 21*x^4 - 7*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A225009 Number of n X 7 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

29, 239, 1163, 4166, 12174, 30738, 69498, 144111, 278707, 508937, 885677, 1479452, 2385644, 3730548, 5678340, 8439021, 12277401, 17523187, 24582239, 33949058, 46220570, 62111270, 82469790, 108296955, 140765391, 181240749, 231304609

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

			Some solutions for n=3:
..0..1..1..0..0..0..0....1..1..1..1..0..0..0....0..0..0..0..0..0..0
..0..1..1..1..1..0..0....1..1..1..1..1..0..0....1..1..0..0..0..0..0
..1..1..1..1..1..1..1....1..1..1..1..1..1..1....1..1..0..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 7 of A225010.

Empirical: a(n) = (4/315)*n^7 + (1/5)*n^6 + (58/45)*n^5 + (35/8)*n^4 + (1507/180)*n^3 + (357/40)*n^2 + (2027/420)*n + 1.
Conjectures from Colin Barker, Sep 05 2018: (Start)
G.f.: x*(29 + 7*x + 63*x^2 - 70*x^3 + 56*x^4 - 28*x^5 + 8*x^6 - x^7) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)

A225011 Number of 4 X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

5, 25, 95, 295, 791, 1897, 4166, 8518, 16414, 30086, 52834, 89402, 146446, 233108, 361711, 548591, 815083, 1188679, 1704377, 2406241, 3349193, 4601059, 6244892, 8381596, 11132876, 14644540, 19090180, 24675260, 31641640, 40272566, 50898157

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

Row 4 of A225010.

Apparently also column 5 of A071920. - R. J. Mathar, May 17 2014

			Some solutions for n=3
..0..1..0....0..0..0....0..0..1....0..0..0....1..0..0....1..0..0....0..0..1
..0..1..1....0..1..0....0..1..1....0..0..0....1..0..0....1..0..0....0..0..1
..1..1..1....0..1..0....1..1..1....0..0..0....1..1..0....1..0..0....0..0..1
..1..1..1....0..1..0....1..1..1....0..0..1....1..1..0....1..1..1....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A071920, A225010.

Empirical: a(n) = (1/40320)*n^8 + (1/1440)*n^7 + (3/320)*n^6 + (5/72)*n^5 + (629/1920)*n^4 + (1279/1440)*n^3 + (16763/10080)*n^2 + (25/24)*n + 1 = 1 + n* (n+1) *(n^6 + 27*n^5 + 351*n^4 + 2449*n^3 + 10760*n^2 + 25052*n + 42000)/40320.

Empirical: G.f.: -x*(x^4 - 5*x^3 + 10*x^2 - 10*x + 5) *(x^4 - 3*x^3 + 4*x^2 - 2*x + 1) / (x-1)^9. - R. J. Mathar, May 17 2014

A225012 Number of 5 X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

6, 36, 161, 581, 1792, 4900, 12174, 27966, 60172, 122464, 237590, 442118, 793092, 1377174, 2322967, 3817351, 6126818, 9624964, 14827487, 22436251, 33394208, 48953224, 70757132, 100942636, 142261016, 198223936, 273277036, 373005396

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

Row 5 of A225010.

Apparently column 6 of A071920. - R. J. Mathar, May 17 2014

			Some solutions for n=3
..0..0..0....0..1..0....0..1..0....1..0..0....0..0..0....0..0..0....0..0..0
..1..0..0....0..1..0....1..1..0....1..1..0....0..0..1....0..0..0....0..1..1
..1..0..0....0..1..0....1..1..1....1..1..0....0..1..1....0..1..0....0..1..1
..1..1..0....0..1..1....1..1..1....1..1..1....0..1..1....0..1..1....1..1..1
..1..1..0....0..1..1....1..1..1....1..1..1....0..1..1....1..1..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/3628800)*n^10 + (1/80640)*n^9 + (1/3780)*n^8 + (19/5760)*n^7 + (4633/172800)*n^6 + (331/2304)*n^5 + (191249/362880)*n^4 + (24421/20160)*n^3 + (10897/5600)*n^2 + (137/120)*n + 1 = 1 + n*(n+1)* (n^8 + 44*n^7 + 916*n^6 + 11054*n^5 + 86239*n^4 + 435086*n^3 + 1477404*n^2 + 2918376*n + 4142880)/ 3628800.

Empirical: G.f.: -x*(x^2 - 3*x + 3) *(x^2 - 2*x + 2) *(x^2 - x + 1) *(x^4 - 4*x^3 + 5*x^2 - 2*x + 1) / (x-1)^11. - R. J. Mathar, May 17 2014

cp's OEIS Frontend

A059259 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-x-x*y-y^2) = 1/((1+y)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...

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A071920 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=0 for all m>=0, read by antidiagonals.

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A071921 Square array giving number of unimodal functions [n]->[m] for n>=0, m>=0, with a(0,m)=1 by definition, read by antidiagonals.

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A239473 Triangle read by rows: signed version of A059260: coefficients for expansion of partial sums of sequences a(n,x) in terms of their binomial transforms (1+a(.,x))^n ; Laguerre polynomial expansion of the truncated exponential.

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A225006 Number of n X n 0..1 arrays with rows unimodal and columns nondecreasing.

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A225007 Number of n X 5 0..1 arrays with rows unimodal and columns nondecreasing.

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A225008 Number of n X 6 0..1 arrays with rows unimodal and columns nondecreasing.

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