A104203 -id:A104203 - cp's OEIS frontend

A159600 E.g.f. C(x) satisfies: C(x) = (1 - 2*S(x)^2)^(1/4), where S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

1, -1, 3, -27, 441, -11529, 442827, -23444883, 1636819569, -145703137041, 16106380394643, -2164638920874507, 347592265948756521, -65724760945840254489, 14454276753061349098587

Offset: 0

Paul D. Hanna, May 07 2009

sign

See A104203 for the expansion of the sine lemniscate function sl(x).
E.g.f. C(x) is an even function; zero terms are omitted.
Radius of convergence is |x| <= r:
r = sqrt(2)*(Pi/2)^(3/2)/gamma(3/4)^2 with
C(r) = gamma(3/4)^2/(Pi/2)^(3/2) where:
r = L/sqrt(2) where L=Lemniscate constant;
r = 1.8540746773013719184338503471952600...
C(r) = 0.76275976350181318806232598096361579...

			E.g.f.: C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+ ...
C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+ ...
C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+ ...
C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+ ...
C(x)^4 + 2*S(x)^2 = 1 where:
S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! + ...
S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
From _Paul D. Hanna_, Jul 29 2011: (Start)
O.g.f.: 1 - x + 3*x^2 - 27*x^3 + 441*x^4 - 11529*x^5 + 442827*x^6 -+ ... + a(n)*x^n + ...
O.g.f.: 1/(1 + x/(1 + 2*x/(1 + 9*x/(1 + 8*x/(1 + 25*x/(1 + 18*x/(1 + 49*x/(1 + 32*x/(1-...))))))))) (continued fraction). (End)

Cf. A159601 (S(x)); A193541, A193544: All of these have the same |a(n)|. - M. F. Hasler, Aug 31 2012

Cf. A129194.

a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ JacobiCN[ x, 1/2], {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, With[ {s = InverseSeries[ Integrate[ Series[(1 - x^4 / 4) ^ (-1/2), {x, 0, m + 1}], x]]}, m! SeriesCoefficient[ Sqrt[ (2 - s^2) / (2 + s^2)], {x, 0, m}]]]]; (* Michael Somos, Apr 25 2011 *)

{a(n)=local(S=x,C);for(i=0,2*n,S=intformal((1-2*S^2+O(x^(2*n+2)))^(3/4))); C=(1-2*S^2)^(1/4) ;(2*n)!*polcoeff(C,2*n)}

{a(n) = my(A, m); if( n<0, 0, m = 2*n; A = serreverse( intformal( (1 - x^4 / 4 + x * O(x^m)) ^ (-1/2))); m! * polcoeff( sqrt( (2 - A^2) / (2 + A^2)), m))}; /* Michael Somos, Apr 25 2011 */

{a(n) = local(C=1+x); for(i=1,n, C = exp( intformal( C * intformal(-1/C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C,n)}
for(n=0,20,print1(a(2*n),", "))

{a(n) = local(C=1+x); for(i=1,n, C = exp( intformal( -1/C * intformal(C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C,n)}
for(n=0,20,print1(a(2*n),", "))

E.g.f. C(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0; Left-shift of the Laplace transform of e.g.f. C(x) equals the Laplace transform of S(x).
E.g.f.: Sum_{k>=0} 2^k * a(k) * x^(2*k) / (2*k)! = cos lemn(x) where cos lemn(x) is the cosine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
O.g.f.: 1/(1 + 1^2*x/(1 + 2^2/2*x/(1 + 3^2*x/(1 + 4^2/2*x/(1 + 5^2*x/(1 + 6^2/2*x/(1 + 7^2*x/(1 + 8^2/2*x/(1+...))))))))) (continued fraction). - Paul D. Hanna, Jul 29 2011
a(n) = (-1)^floor(n/2) * A193544(n) = (-1)^ceiling(n/2) * A193544(n) = -A159601(n). - M. F. Hasler, Aug 31 2012
G.f.: 1/Q(0) where p=1/2, Q(k) = 1 + x*(2*k+1)^2/( 1 + p*x*(2*k+2)^2/Q(k+1) ); (continued fraction due to Stieltjes T.J.). - Sergei N. Gladkovskii, Mar 22 2013
From Paul D. Hanna, Jun 02 2015: (Start)
E.g.f. C(x) satisfies:
(1) C(x) = exp( Integral C(x) * Integral -1/C(x)^3 dx dx ).
(2) C(x) = exp( Integral -1/C(x) * Integral C(x)^3 dx dx ).
(End)
G.f.: 1 / (1 + b(1)*x / (1 + b(2)*x / (1 + b(3)*x / ... ))) where b = A129194. - Michael Somos, Jan 03 2013

A159601 E.g.f. S(x) satisfies: S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0.

Original entry on oeis.org

1, -3, 27, -441, 11529, -442827, 23444883, -1636819569, 145703137041, -16106380394643, 2164638920874507, -347592265948756521, 65724760945840254489, -14454276753061349098587

Offset: 1

Paul D. Hanna, May 08 2009

sign

E.g.f. S(x) is an odd function; zero terms are omitted.
Apart from signs and initial term, same as A159600.
Radius of convergence of S(x) is |x| < r where:
r = (1/2)*Pi^(3/2)/gamma(3/4)^2 ;
r = L/sqrt(2) where L=Lemniscate constant ;
r = 1.8540746773013719184338503471952600...
Although S(x) diverges at |x|=r, the power series expansion:
C(x) = [1 - 2*S(x)^2]^(1/4) converges to
C(r) = gamma(3/4)^2/(Pi/2)^(3/2) = 0.7627597635...

			E.g.f: S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...
S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
C(x)^4 + 2*S(x)^2 = 1 where:
C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+...
C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+...
C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+...
C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+...
1/C(x) = C(i*x) = 1 + x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! +...
log(C(x)) = -x^2/2! - 12*x^6/6! - 3024*x^10/10! - 4390848*x^14/14! -...
Coefficients in log(C(x)) are given by A104203 (ignoring signs).

Cf. A159600 (C(x)), A104203 (unsigned e.g.f. = S(x)/C(x)).

{a(n)=local(S=x);for(i=0,2*n,S=intformal((1-2*S^2+O(x^(2*n)))^(3/4)));(2*n-1)!*polcoeff(S,2*n-1)}

E.g.f. S(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where
S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.
E.g.f. S(x) satisfies: S(x)/C(x) = e.g.f. of unsigned A104203 where C(x)^4 + 2*S(x)^2 = 1.
a(n) = -A159600(n), n>0. - M. F. Hasler, Aug 31 2012
G.f.: (1- 1/Q(0))/x, where Q(k) = 1 + x*(2*k+1)^2/(1 + 2*x*(k+1)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 30 2013

A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -3, -12, -27, 0, 441, 3024, 11529, 0, -442827, -4390848, -23444883, 0, 1636819569, 21224560896, 145703137041, 0, -16106380394643, -257991277243392, -2164638920874507, 0, 347592265948756521

Offset: 0

Peter Luschny, Jul 26 2011

sign

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Eric W. Weisstein, Jacobi Elliptic Functions.

Cf. A060627, A060628, A104203, A159600, A193541, A193544.

A190904 := proc(n) option remember; `if`(n=0,1,add(((1-irem(k,2))*(-1)^ iquo(k,2))*binomial(n-1,k)*A190904(n-1-k)*A190904(k),k=0..n-1)) end:

a[0] = 1;
a[n_] := a[n] =
  Sum[Mod[(k+1)^3, 4, -1] Binomial[n-1, k] a[n-k-1] a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 24 2019 *)

Let F(n,x) = Sum_{k=0..n-1} cos(Pi*k*x)*binomial(n-1,k)*F(n-1-k,x)* F(k,x), then
F(n, 0)   = n! = A000142(n),
F(n, 1/2) = a(n),
F(n, 1)   = 2^n*Euler_{n}(1) = A_{n}(-1) = A155585(n).
a(2*n) = A159601(n)*(-1)^floor((n-1)/2).
a(2*n+1) = A104203(2*n+1).
From Peter Bala, Aug 25 2011: (Start)
The sequence entries may be calculated as follows: Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. The coefficients in the expansion of D^n[f](x) in powers of f(x) can be found in A145271. Then we have
a(2*n) = D^(2*n)[sqrt(1+sin^2(x))](0)
a(2*n+1) = D^(2*n)[sqrt(1-x^4)](0).
The generating function involves the Jacobian elliptic functions. Define E(u,k) := cn(i*u,k)-i*sn(i*u,k) = 1+u+u^2/2!+(1+k^2)*u^3/3!+(1+4*k^2)*u^4/4!+..., where cn(u,k) and sn(u,k) are Jacobian elliptic functions of modulus k (see A060627 and A060628). Then the e.g.f. A(u) for this sequence is
A(u) := E(u,i) = 1+u+u^2/2!-3*u^4/4!-12*u^5/5!-27*u^6/6!+....
Proof: Using well-known properties of the Jacobian elliptic functions (see for example Abramowitz and Stegun, Chapter 16) we find the generating function A(u) satisfies the differential equation
(d/du)A(u) = dn(i*u,i)*A(u) = 1/2*(A(i*u)+A(-i*u))*A(u), which leads to a recurrence for the coefficients of A(u):
a(n+1) = sum{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*a(2*k)*a(n-2*k) with a(0) = 1. This recurrence is equivalent to the defining recurrence for this sequence given above.
End proof.
The generating function A(u) satisfies 1/A(u) = A(-u).
Compare entries of this sequence with those of A104203, A159600, A193541 and A193544.
(End)

A242240 Expansion of Jacobi sn(x, 1/2) / cd(x, 1/2).

Original entry on oeis.org

0, 1, 0, 0, 0, 12, 0, 0, 0, 3024, 0, 0, 0, 4390848, 0, 0, 0, 21224560896, 0, 0, 0, 257991277243392, 0, 0, 0, 6628234834692624384, 0, 0, 0, 319729080846260095008768, 0, 0, 0, 26571747463798134334265819136, 0, 0, 0, 3564202847752289659513902717468672, 0, 0, 0

Offset: 0

Michael Somos, May 09 2014

nonn

			G.f. = x + 12*x^5 + 3024*x^9 + 4390848*x^13 + 21224560896*x^17 + ...

Cf. A104203.

a[ n_] := If[ n<0, 0, n! SeriesCoefficient[ JacobiSN[x, 1/2] / JacobiCD[x, 1/2], {x, 0, n}]];

{a(n) = if( n<0, 0, n! * polcoeff( serreverse( intformal( (1 + x^4 + x * O(x^n))^(-1/2))), n))};

a(n) = |A104203(n)|.
E.g.f.: sn(x, 1/2) / cd(x, 1/2).
E.g.f. A(x) satisfies A(x)^2 = sinh(2 * Integral A(x) dx). - Michael Somos, Jun 17 2017

A261000 Unordered even-degree bilabeled increasing trees on 2n+1 nodes.

Original entry on oeis.org

1, 3, 189, 68607, 82908441, 251944606683, 1618221395188629, 19514714407120367127, 405452689572115086887601, 13596354857453497541480646963, 699110237190377161907394095173869, 52888313306236766686682435536884784047

Offset: 0

N. J. A. Sloane, Aug 09 2015

nonn

Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014. See p. 18

Closely related to A104203.

Cf. A258659.

A261000aer := proc(n)
    option remember;
    local a,nloc,j,k,l;
    if n = 1 then
        1;
    else
        nloc := n-2 ;
        a :=0 ;
        for j from 0 to nloc-1 do
            for k from 0 to nloc-1-j do
                l := nloc-1-j-k ;
                if l >= 0 then
                    a := a+procname(j+1)*procname(k+1)*procname(l+1) * (2*nloc+1)!/(2*j+1)!/(2*k+1)!/(2*l+1)! ;
                end if;
            end do:
        end do:
        %/2 ;
    end if;
end proc:
A261000 := proc(n)
    A261000aer(2*n+1) ;
end proc:
seq(A261000(n),n=0..15) ; # R. J. Mathar, Aug 18 2015

terms = 12; nmax = 4 terms; A = 1; Do[A = Exp[Integrate[A^(1/2)*Integrate[1/A^(3/2), x], x] + O[x]^nmax], nmax]; A258659 = CoefficientList[A, x^2]*Range[0, nmax - 2, 2]!;
a[n_] := A258659[[2 n + 1]];
Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Nov 27 2017 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ JacobiSD[x, 1/2], {x, 0, 4*n+1}]]; (* Michael Somos, Sep 03 2022 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[ x*Sqrt[1/x^2 / WeierstrassP[x, {1, 0}]], {x, 0, 4*n+1}]]; (* Michael Somos, Jul 02 2024 *)
a[ n_] := If[ n<0, 0, (-1)^n * (4*n+1)! * SeriesCoefficient[
InverseSeries[ Series[ x * Hypergeometric2F1[1/4, 1/2, 5/4, x^4/4], {x, 0, 4*n+1}]], {x, 0, 4*n+1}]]; (* Michael Somos, Jul 02 2024 *)

{a(n) = if( n<0, 0, my(m = 4*n + 1); m! * polcoeff( serreverse( intformal( 1 / sqrt(1 + x^4/4 + x * O(x^m)) ) ), m))}; /* Michael Somos, Jun 17 2017 */

Kuba et al. (2014) gives a recurrence (see Theorem 7).

a(n) = A258659(2*n). - Michael Somos, Jun 17 2017

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(x,q) = Sum_{n>=0} sum_{k=0..n(n+1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 4, 1, 35, 91, 85, 20, 16, 24, 1, 84, 966, 1324, 2737, 632, 1200, 288, 384, 128, 192, 1, 165, 5082, 26818, 50941, 64329, 69816, 49872, 27568, 19776, 16448, 10176, 5760, 3840, 1280, 1920, 1, 286, 18447, 279136, 954239, 2550054, 2455233, 4013788, 2929104, 3264864, 1176640, 1815552, 834752, 731136, 394752, 491264, 141056, 164352, 69120, 46080, 15360, 23040, 1, 455, 53053, 1780207, 15627183, 51699869, 128611679, 187372653, 213804652, 257006976, 245800968, 195109120, 161177792, 123750592, 83792000, 69316224, 52893696, 35140992, 28215808, 18433536, 12687360, 8411648, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560

Offset: 0

Paul D. Hanna, Dec 16 2018

Compare to Jacobi's elliptic function sn(x,k) = Integral cn(x,k)*dn(x,k) dx such that cn(x,k)^2 + sn(x,k)^2 = 1 and dn(x,k)^2 + k^2*sn(x,k)^2 = 1.
Right border equals A002866.
Row sums equal the tangent numbers (A000182).
Last n terms in row n of this triangle and of triangle A322218 are equal for n>0.

			E.g.f. S(x,q) = Sum_{n>=0} sum_{k=0..n*(n+1)/2} T(n,k)*x^(2*n+1)*q^(2*k)/(2*n+1)! starts
S(x,q) = x + (q^2 + 1)*x^3/3! + (4*q^6 + q^4 + 10*q^2 + 1)*x^5/5! + (24*q^12 + 16*q^10 + 20*q^8 + 85*q^6 + 91*q^4 + 35*q^2 + 1)*x^7/7! + (192*q^20 + 128*q^18 + 384*q^16 + 288*q^14 + 1200*q^12 + 632*q^10 + 2737*q^8 + 1324*q^6 + 966*q^4 + 84*q^2 + 1)*x^9/9! + (1920*q^30 + 1280*q^28 + 3840*q^26 + 5760*q^24 + 10176*q^22 + 16448*q^20 + 19776*q^18 + 27568*q^16 + 49872*q^14 + 69816*q^12 + 64329*q^10 + 50941*q^8 + 26818*q^6 + 5082*q^4 + 165*q^2 + 1)*x^11/11! + ...
such that S(x,q) = sinh( Integral C(q*x,q) dx ) and C(x,q)^2 = 1 + S(x,q)^2.
This irregular triangle of coefficients T(n,k) of x^(2*n+1)*q^(2*k)/(2*n+1)! in S(x,q) begins:
1;
1, 1;
1, 10, 1, 4;
1, 35, 91, 85, 20, 16, 24;
1, 84, 966, 1324, 2737, 632, 1200, 288, 384, 128, 192;
1, 165, 5082, 26818, 50941, 64329, 69816, 49872, 27568, 19776, 16448, 10176, 5760, 3840, 1280, 1920;
1, 286, 18447, 279136, 954239, 2550054, 2455233, 4013788, 2929104, 3264864, 1176640, 1815552, 834752, 731136, 394752, 491264, 141056, 164352, 69120, 46080, 15360, 23040;
1, 455, 53053, 1780207, 15627183, 51699869, 128611679, 187372653, 213804652, 257006976, 245800968, 195109120, 161177792, 123750592, 83792000, 69316224, 52893696, 35140992, 28215808, 18433536, 12687360, 8411648, 5945856, 2673664, 2300928, 967680, 645120, 215040, 322560;
1, 680, 129948, 8212360, 163115238, 1001312104, 3705217660, 7815443320, 15434182497, 17298854576, 23429393056, 21144463040, 25624143104, 18454639872, 18756800128, 12036914176, 12076688384, 7122865152, 7609525248, 4420732928, 4042876928, 2553473024, 2465701888, 1353586688, 1234018304, 619528192, 587358208, 311279616, 255467520, 117383168, 108036096, 42778624, 36814848, 15482880, 10321920, 3440640, 5160960; ...
RELATED SERIES.
C(x,q) = 1 + x^2/2! + (4*q^2 + 1)*x^4/4! + (24*q^6 + 16*q^4 + 20*q^2 + 1)*x^6/6! + (192*q^12 + 128*q^10 + 384*q^8 + 288*q^6 + 336*q^4 + 56*q^2 + 1)*x^8/8! + (1920*q^20 + 1280*q^18 + 3840*q^16 + 5760*q^14 + 10176*q^12 + 5888*q^10 + 12736*q^8 + 6448*q^6 + 2352*q^4 + 120*q^2 + 1)*x^10/10! + (23040*q^30 + 15360*q^28 + 46080*q^26 + 69120*q^24 + 164352*q^22 + 141056*q^20 + 341504*q^18 + 294912*q^16 + 431616*q^14 + 385472*q^12 + 472704*q^10 + 214016*q^8 + 93280*q^6 + 10032*q^4 + 220*q^2 + 1)*x^12/12! + ...
such that C(x,q) = cosh( Integral C(q*x,q) dx ).

Paul D. Hanna, Table of n, a(n) for n = 0..4990, as a flattened irregular triangle read by rows 0..30.

Cf. A322218 (C(x,q)), A000182 (row sums), A104203, A002866.

rows = 8; m = 2 rows; s[x_, ] = x; c[, ] = 1; Do[s[x, q_] = Integrate[c[x, q] c[q x, q] + O[x]^m // Normal, x]; c[x_, q_] = 1 + Integrate[s[x, q] c[q x, q] + O[x]^m // Normal, x], {m}];
CoefficientList[#, q^2]& /@ (CoefficientList[s[x, q], x] Range[0, m-1]!) // DeleteCases[#, {}]& // Flatten (* Jean-François Alcover, Dec 17 2018 *)

{T(n,k) = my(S=x,C=1); for(i=1,2*n,
S = intformal(C*subst(C,x,q*x) +O(x^(2*n+1)));
C = 1 + intformal(S*subst(C,x,q*x)));
(2*n+1)!*polcoeff( polcoeff(S,2*n+1,x),2*k,q)}
for(n=0,10, for(k=0,n*(n+1)/2, print1( T(n,k),", "));print(""))

E.g.f. S(x,q) and related series C(x,q) satisfy:
(1) C(x,q)^2 - S(x,q)^2 = 1.
(2) C(x,q) = 1 + Integral S(x,q) * C(q*x,q) dx.
(3) S(x,q) = Integral C(x,q) * C(q*x,q) dx.
(4a) C(x,q) + S(x,q) = exp( Integral C(q*x,q) dx ).
(4b) C(x,q) = cosh( Integral C(q*x,q) dx ).
(4c) S(x,q) = sinh( Integral C(q*x,q) dx ).
(5) C(q*x,q) = 1 + q * Integral S(q*x,q) * C(q^2*x,q) dx.
(6) S(q*x,q) = q * Integral C(q*x,q) * C(q^2*x,q) dx.
(7a) C(q*x,q) + S(q*x,q) = exp( q * Integral C(q^2*x,q) dx ).
(7b) C(q*x,q) = cosh( q * Integral C(q^2*x,q) dx ).
(7c) S(q*x,q) = sinh( q * Integral C(q^2*x,q) dx ).
PARTICULAR ARGUMENTS.
S(x,q=0) = sinh(x).
S(x,q=1) = tan(x).
S(x,q=i) = -i * sl(i*x), where sl(x) is the sine lemniscate function (A104203).
FORMULAS FOR TERMS.
T(n, n*(n+1)/2) = 2^(n-1)*n! for n >= 1.
T(n, n*(n+1)/2 - k) = A322218(n, n*(n-1)/2 - k) for k = 0..n-1, n > 0.
Sum_{k=0..n*(n+1)/2} T(n,k) = A000182(n+1) for n >= 0.
Sum_{k=0..n*(n+1)/2} T(n,k)*(-1)^k = A104203(2*n+1) for n >= 0.

cp's OEIS Frontend

A283831 Take every fourth term of A104203.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A159600 E.g.f. C(x) satisfies: C(x) = (1 - 2*S(x)^2)^(1/4), where S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A159601 E.g.f. S(x) satisfies: S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A190904 a(n) = Sum_{k=0..n-1} cos(Pi*k/2)*binomial(n-1,k)*a(n-1-k)*a(k) for n > 0, a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A242240 Expansion of Jacobi sn(x, 1/2) / cd(x, 1/2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A261000 Unordered even-degree bilabeled increasing trees on 2n+1 nodes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(q*x,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(x,q) = Sum_{n>=0} sum_{k=0..n*(n+1)/2} T(n,k)*x^n*y^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

A322219 E.g.f.: S(x,q) = Integral C(x,q) * C(qx,q) dx, such that C(x,q)^2 - S(x,q)^2 = 1, where S(x,q) = Sum_{n>=0} sum_{k=0..n(n+1)/2} T(n,k)x^ny^k/n!, as an irregular triangle of coefficients T(n,k) read by rows.