A210657 -id:A210657 - cp's OEIS frontend

A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.

1, 0, 1, 0, 0, 1, 3, 1, 0, 0, 3, 15, 25, 15, 3, 0, 0, 17, 119, 329, 455, 329, 119, 17, 0, 0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0, 0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0, 0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0, 0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0

Offset: 0

Paul D. Hanna, May 01 2016

Row sums equal the Euler numbers, A000364.
Column 1 equals A110501, the unsigned Genocchi numbers of first kind.
Main diagonal equals A272482, where A272482(n) = A005799(n)/2^n * (2*n)!/(n!)^2.
Sum_{k=0..2*n} (-1)^k*T(n,k) = (-1)^n.
Sum_{k=0..2*n} (-2)^k*T(n,k) = 2*(-1)^n for n>0.
Sum_{k=0..2*n} 2^k*T(n,k) = (-1)^n*A210657(n).
Sum_{k=0..2*n} 3^k*T(n,k) = A000281(n).
Sum_{k=0..2*n} 4^k*T(n,k) = A272158(n).
Sum_{k=0..2*n} 2^k*3^(2*n-k)*T(n,k) = A272467(n).

			E.g.f.: A(x,y) = 1 + x^2*(y)/2! + x^4*(y + 3*y^2 + y^3)/4! +
x^6*(3*y + 15*y^2 + 25*y^3 + 15*y^4 + 3*y^5)/6! +
x^8*(17*y + 119*y^2 + 329*y^3 + 455*y^4 + 329*y^5 + 119*y^6 + 17*y^7)/8! +
x^10*(155*y + 1395*y^2 + 5325*y^3 + 11235*y^4 + 14301*y^5 + 11235*y^6 + 5325*y^7 + 1395*y^8 + 155*y^9)/10! +
x^12*(2073*y + 22803*y^2 + 110605*y^3 + 311355*y^4 + 563013*y^5 + 683067*y^6 + 563013*y^7 + 311355*y^8 + 110605*y^9 + 22803*y^10 + 2073*y^11)/12! +...
where A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2).
This triangle of coefficients of x^(2*n)*y^k/(2*n)!, k=0..2*n, begins:
[1];
[0, 1, 0];
[0, 1, 3, 1, 0];
[0, 3, 15, 25, 15, 3, 0];
[0, 17, 119, 329, 455, 329, 119, 17, 0];
[0, 155, 1395, 5325, 11235, 14301, 11235, 5325, 1395, 155, 0];
[0, 2073, 22803, 110605, 311355, 563013, 683067, 563013, 311355, 110605, 22803, 2073, 0];
[0, 38227, 496951, 2918825, 10241231, 23904881, 39102063, 45956625, 39102063, 23904881, 10241231, 2918825, 496951, 38227, 0];
[0, 929569, 13943535, 96075665, 403075855, 1150348017, 2362479119, 3600524785, 4136759055, 3600524785, 2362479119, 1150348017, 403075855, 96075665, 13943535, 929569, 0]; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1088, of flattened triangle for rows 0..32.

Cf. A000364, A110501, A272482, A210657, A000281, A272158, A272467.

{T(n,k) = my(X=x+x*O(x^(2*n))); (2*n)!*polcoeff(polcoeff( cos((X-x*y)/2)/cos((X+x*y)/2), 2*n,x), k,y)}
for(n=0,10, for(k=0,2*n, print1(T(n,k),", "));print(""))

E.g.f.: A(x,y) = (cos(x) + cos(x*y)) / (1 + cos(x + x*y)).
E.g.f.: A(x,y) = (sin(x) + sin(x*y)) / sin(x + x*y).
E.g.f.: A(x,y) = (exp(i*x) + exp(i*x*y)) / (1 + exp(i*(x + x*y))), where i^2 = -1.
O.g.f.: 1/(1 - 1*y*x/(1 - (1+y)^2*x/(1 - (1+2*y)*(2+1*y)*x/(1 - (2+2*y)^2*x/(1 - (2+3*y)*(3+2*y)*x/(1 - (3+3*y)^2*x/(1 - (3+4*y)*(4+3*y)*x/(1 - (4+4*y)^2*x/(1 - (4+5*y)*(5+4*y)*x/(1 - (5+5*y)^2*x/(1 - ...))))))))))), a continued fraction.

A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 486, 21944, 1591210, 169207092, 24808395262, 4796420822384, 1182349445882706, 361939981107422060, 134705596642758848806, 59900689507397744253096, 31365504832631796986962426, 19102102945852191813235300004, 13387748268024668296590660222030

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A000111, A000364, A009214, A205571, A210657, A302397, A352251, A352252.

nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k+1) * binomial(2*n,2*k) * k * a(n-k).

A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.

The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).

			[0] [      1]
[1] [     -1,         1]
[2] [      5,       -11,        6]
[3] [    -61,       211,     -240,        90]
[4] [   1385,     -6551,    11466,     -8820,     2520]
[5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.

Row sums are A000007, alternating row sums are A210657.
Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);

Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0,k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)

def EW(m, n):
    @cached_function
    def S(m, n):
        R. = ZZ[]
        if n == 0: return R(1)
        return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
    s = S(m, n).list()
    c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
        sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
    return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])

Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:

T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).

A350745 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k loops, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 84, 32, 1, 1, 80, 460, 460, 80, 1, 1, 192, 2190, 4600, 2190, 192, 1, 1, 448, 9534, 37310, 37310, 9534, 448, 1, 1, 1024, 39032, 264208, 483140, 264208, 39032, 1024, 1, 1, 2304, 152856, 1702344, 5229756, 5229756, 1702344, 152856, 2304, 1

Offset: 0

David Galvin, Jan 13 2022

Loop-threshold graphs are constructed from either a single unlooped vertex or a single looped vertex by iteratively adding isolated vertices (adjacent to nothing previously added) and looped dominating vertices (looped, and adjacent to everything previously added).

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   12,     12,       1;
  1,   32,     84,      32,       1;
  1,   80,    460,     460,      80,       1;
  1,  192,   2190,    4600,    2190,     192,       1;
  1,  448,   9534,   37310,   37310,    9534,     448,      1;
  1, 1024,  39032,  264208,  483140,  264208,   39032,   1024,    1;
  1, 2304, 152856, 1702344, 5229756, 5229756, 1702344, 152856, 2304, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).
D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021.

Row sums are A000629.
Columns k=0..1 give: A000012, A001787,
Cf. A210657.

T[n_, 0] := T[n, 0] = 1; T[n_, k_] := T[n, k] = Binomial[n, k]*Sum[Factorial[l]*StirlingS2[k, l]*(Factorial[l - 1]*StirlingS2[n - k, l - 1] + 2*Factorial[l]*StirlingS2[n - k, l] + Factorial[l + 1]*StirlingS2[n - k, l + 1]), {l, 1, n + 1}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]

T(n,k) = if(k==0, 1, binomial(n,k) * sum(j=1, n, j!*stirling(k,j,2) * ((j-1)! * stirling(n-k,j-1,2) + 2*j!*stirling(n-k,j,2) + (j+1)!*stirling(n-k,j+1,2)))) \\ Andrew Howroyd, May 06 2023

T(n,0) = 1; T(n,k) = binomial(n,k) * Sum_{j=1..n} j!*Stirling2(k,j) * ((j-1)! * Stirling2(n-k,j-1) + 2*j!*Stirling2(n-k,j) + (j+1)!*Stirling2(n-k,j+1)).
T(n,k) = T(n,n-k).
Sum_{k=0..2*n} (-1)^k * T(2*n,k) = A210657(n). - Alois P. Heinz, Feb 01 2022

A352277 a(0) = 1; a(n) = -2 * Sum_{k=1..n} binomial(2n-1,2k-1) * a(n-k).

Original entry on oeis.org

1, -2, 10, -62, 250, 3538, -109430, 376738, 64406170, -1496149262, -66387156950, 4120939699138, 114360544465210, -16447057086702062, -315993884108535350, 99921676927889325538, 1478937314465295441370, -907773678752741550637262, -14225447208333541085396630

Offset: 0

Ilya Gutkovskiy, Mar 10 2022

sign

Cf. A000807, A210657, A260884, A352278.

a[0] = 1; a[n_] := a[n] = -2 Sum[Binomial[2 n - 1, 2 k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 36; Take[CoefficientList[Series[Exp[2 (1 - Cosh[x])], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]

E.g.f.: exp( 2 * (1 - cosh(x)) ) (even powers only).

cp's OEIS Frontend

A272481 E.g.f. A(x,y) = cos((x - x*y)/2) / cos((x + x*y)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2*n)*y^k/n! in A(x,y), for k=0..2*n.

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A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

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A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

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A350745 Triangle read by rows: T(n,k) is the number of labeled loop-threshold graphs on vertex set [n] with k loops, for n >= 0 and 0 <= k <= n.

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A352277 a(0) = 1; a(n) = -2 * Sum_{k=1..n} binomial(2*n-1,2*k-1) * a(n-k).

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A272481 E.g.f. A(x,y) = cos((x - xy)/2) / cos((x + xy)/2) represented as a triangle, read by rows, where row n lists of coefficients T(n,k) of x^(2n)y^k/n! in A(x,y), for k=0..2*n.

A352277 a(0) = 1; a(n) = -2 * Sum_{k=1..n} binomial(2n-1,2k-1) * a(n-k).