A002105 -id:A002105 - cp's OEIS frontend

A218221 G.f.: 1/(1-x/(1-4x/(1-10x/(1-20x/(1-35x/(1-56*x/(1-...))))))), a continued fraction.

1, 1, 5, 65, 1725, 81225, 6181125, 710984625, 117537778125, 26848583825625, 8210318193703125, 3275053250628290625, 1667519951972905828125, 1063947235962694359515625, 837322677987349287566953125, 801714108831393845941434140625

Offset: 0

Paul D. Hanna, Oct 23 2012

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1725*x^4 + 81225*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..198

Cf. A000292, A002105, A216966.

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Binomial[Range[nmax + 1]+2,3]*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)
eulerCF[f_, len_] := Module[{g}, g[len - 1] = 1;
g[k_] := g[k] = 1 - f[k]/(f[k] - 2/g[k + 1]); CoefficientList[g[0] + O[x]^len, x]];
A218221List[len_] := eulerCF[(1/3) x (# + 1) (# + 2) (# + 3) &, len];
A218221List[16] (* Peter Luschny, Aug 09 2019 *)

{a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)*(n-k+2)*(n-k+3)/6*x*CF)); polcoeff(CF, n)}
for(n=0,20,print1(a(n),", "))

a(n) == 0 (mod 5) for n>1.
a(n) == 0 (mod 3) for n>3.
G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)*(k+3)/(x*(k+1)*(k+2)*(k+3) - 6/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013
a(n) ~ c * d^n * (n!)^3 * n^(5/2), where d = 32*sqrt(3)*Pi^3 / Gamma(1/3)^9 = 0.241821816937867322064737317131437... and c = 2^(15/2) * 3^(11/4) * Pi^4 / Gamma(1/3)^(27/2) = 0.60382458700655692263505976... - Vaclav Kotesovec, Aug 25 2017, updated Mar 16 2024

A285484 G.f.: 1/(1 + x/(1 + x^3/(1 + x^6/(1 + x^10/(1 + x^15/(1 + ... + x^(k*(k+1)/2)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -6, 9, -13, 18, -26, 38, -54, 77, -111, 160, -229, 328, -472, 679, -974, 1398, -2010, 2888, -4146, 5954, -8555, 12289, -17647, 25346, -36410, 52297, -75109, 107881, -154961, 222574, -319679, 459167, -659528, 947295, -1360612, 1954295, -2807031, 4031809, -5790982

Offset: 0

Ilya Gutkovskiy, Apr 19 2017

sign

			G.f.: A(x) = 1 - x + x^2 - x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 9*x^8 - 13*x^9 + ...

Cf. A000217, A002105, A206740, A285408.

nmax = 45; CoefficientList[Series[1/(1 + ContinuedFractionK[x^(k (k + 1)/2), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ (-1)^n * c * d^n, where d = 1.43632929358192465555987661527... and c = 0.4856490524128736949896673... - Vaclav Kotesovec, Aug 26 2017

A365674 Triangle read by rows. T(n, k) = ((n - k + 1)(n - k + 2)/2) T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 10, 34, 34, 1, 20, 154, 496, 496, 1, 35, 504, 3520, 11056, 11056, 1, 56, 1344, 16960, 112816, 349504, 349504, 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104, 1, 120, 6468, 199408, 3739736, 42238560, 268304464, 819786496, 819786496

Offset: 0

Peter Luschny, Sep 30 2023

This triangle is associated to the case n = 3 of A365673 and has as weight function the triangular numbers A000217. The numbers on its main diagonal are the reduced tangent numbers A002105. For details see A365673.

			[0] 1;
[1] 1,  1;
[2] 1,  4,    4;
[3] 1, 10,   34,    34;
[4] 1, 20,  154,   496,    496;
[5] 1, 35,  504,  3520,  11056,   11056;
[6] 1, 56, 1344, 16960, 112816,  349504,   349504;
[7] 1, 84, 3108, 63580, 748616, 4841200, 14873104, 14873104;

Cf. A002105 (main diagonal), A365673 (case n=3), A000217 (weight).

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k - 1) else ((n - k + 1)*(n - k + 2)/2) * T( n, k - 1) + T( n - 1, k) fi fi end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

A094346 Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 3, 8, 6, 0, 17, 54, 60, 24, 0, 155, 556, 762, 480, 120, 0, 2073, 8146, 12840, 10248, 4200, 720, 0, 38227, 161424, 282078, 263040, 139440, 40320, 5040, 0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320

Offset: 0

Philippe Deléham, Jun 08 2004, Jun 13 2007

Diagonals: A000007, A001469, A005440; A000182, A005990. Row sums: A001469.

			Triangle begins:
1;
0, 1;
0, 1, 2;
0, 3, 8, 6;
0, 17, 54, 60, 24;
0, 155, 556, 762, 480, 120;
0, 2073, 8146, 12840, 10248, 4200, 720;
0, 38227, 161424, 282078, 263040, 139440, 40320, 5040;
0, 929569, 4163438, 7886580, 8240952, 5170800, 1965600, 423360, 40320; ...

D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
J. M. Gandhi, Research Problems: A Conjectured Representation of Genocchi Numbers, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
Andrei K. Svinin, Tuenter polynomials and a Catalan triangle, arXiv:1603.05748 [math.CO], 2016. (Has a signed version of this triangle, see p. 1).

Cf. A094665, A083061.

G[_, 1] = 1;
G[x_, n_] := G[x, n] = (x+1)^2 G[x+1, n-1] - x^2 G[x, n-1] // Expand;
row[0] = {1};
row[n_] := CoefficientList[x G[x, n], x];
Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Aug 17 2018 *)

{T(n, k) = local( A = x); if( k<0 || k>n, 0, for( j = 1, n, A = x^2 * ( subst(A, x, x+1) - A)); polcoeff( A, k+1))} /* Michael Somos, Apr 10 2011 */

For n>=1, Sum_{k =1..n} T(n, k)*x^(k-1) = G(x, n), n-th Gandhi polynomial; the Gandhi polynomials are defined by G(x, n) = (x+1)^2*G(x+1, n-1) - x^2*G(x, n-1), G(x, 1) = 1. Sum_{k =0..n} T(n, k)*2^(2n-k) = A000182(n+1), tangent numbers. Sum_{k =0..n} T(n, k) = A001469(n+1), Genocchi numbers of first kind.

Sum_{k = 0..n} T(n, k)*2^(n-k) = A002105(n+1). - Philippe Deléham, Jun 10 2004

A096078 Triangle read by rows: T(n,k) = (k+1)T(n-1,k) + (n-k+1)T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 11, 34, 34, 1, 26, 180, 496, 496, 1, 57, 768, 4288, 11056, 11056, 1, 120, 2904, 28768, 141584, 349504, 349504, 1, 247, 10194, 166042, 1372088, 6213288, 14873104, 14873104, 1, 502, 34096, 868744, 11204160, 82096368, 350400832

Offset: 0

Paul Boddington, Jul 22 2004

			Table begins:
  1
  1 1
  1 4 4
  1 11 34 34
  1 26 180 496 496
  1 57 768 4288 11056 11056

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Cf. A000295, A002105.

T[n_, 0] := 1; T[n_, 1] := 2^(n+1) - n - 2; T[n_, n_] := 2^(n+1)*(2^(2n+2) - 1)*Abs[ BernoulliB[2n + 2]]/ (n + 1); T[n_, k_] := (k + 1)T[n - 1, k] + (n - k + 1)T[n, k - 1]; Flatten[ Table[ T[n, k], {n, 0, 8}, {k, 0, n}]] (* Robert G. Wilson v, Jul 23 2004 *)

T(n-1, 1) given by Eulerian numbers, 2^n - n - 1 (A000295). T(n-1, n-1) given by 2^n*(2^{2n} - 1)*|B_{2n}|/n, B_n = Bernoulli numbers (A002105).

Edited and extended by Robert G. Wilson v, Jul 23 2004

A131638 Increasing binary trees having exactly two vertices with outdegree 1.

Original entry on oeis.org

1, 11, 180, 4288, 141584, 6213288, 350400832, 24718075136, 2133652515072, 221311262045440, 27166907582280704, 3895974311462313984, 645512064907811491840, 122381396964887716078592, 26325690425815766552887296, 6377608610246241663568248832

Offset: 1

Wenjin Woan, Oct 03 2007

nonn

M. P. Develin and S. P. Sullivant, Markov Bases of Binary Graph Models, Annals of Combinatorics 7 (2003) 441-466.
Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., vol. 10, 1989, pp. 369-374.

Cf. A002105, A094503.

Table[n!*SeriesCoefficient[1/2*(-((x*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]) /Sqrt[2]) + 3*Sec[x/Sqrt[2]]^2 *Tan[x/Sqrt[2]]^2), {x, 0, n}], {n, 2, 40, 2}] (* Vaclav Kotesovec after Michel Marcus, Sep 25 2013 *)

lista(m) = { default(realprecision, 30); x = y + O(y^m); egf = (3*tan(x/sqrt(2))^2/cos(x/sqrt(2))^2-x*tan(x/sqrt(2))/(sqrt(2)*cos(x/sqrt(2))^2))/2; forstep (n=2, m, 2, print1(round(n!*polcoeff(egf, n, y)), ", "));}  \\ Michel Marcus, Mar 03 2013

E.g.f.: (3*sec(x/sqrt(2))^2*tan(x/sqrt(2))^2-x*sec(x/sqrt(2))^2*tan(x/sqrt(2))/(sqrt(2)))/2. - Michel Marcus, Mar 03 2013
a(n) ~ (2*n)! * 2^(n+6)*n^3/Pi^(2*n+4). - Vaclav Kotesovec, Sep 25 2013
From Klaus K Haverkamp, Jul 02 2023: (Start)
a(n) = (A002105(n+2) - (n+1)*A002105(n+1))/2.
a(n) = A094503(2n+1,n). (End)

More terms from Michel Marcus, Mar 03 2013

A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

Original entry on oeis.org

1, 7, 3, 127, 123, 30, 4369, 6822, 3579, 630, 243649, 532542, 439899, 162630, 22680, 20036983, 56717781, 64697499, 37155267, 10735470, 1247400, 2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200, 343141433761, 1427877062076, 2563294235106, 2572662311496, 1558544277681, 569674791180, 116270210700, 10216206000

Offset: 1

Matthew Miller, Dec 11 2018

It appears that the first column (7, 127, 4369, ...) is from the sequence A002067.
It appears that the diagonal (3, 30, 630, ...) is from the sequence A007019.
It appears as though the unsigned row sum (10, 280, 15400, ...) is from the sequence A025035.
It appears as though the alternating sign row sum (sum(7, -3) = 4, sum(-127, 123, -30) = -34, ...) is from the sequence A002105.
This triangular array arises as the coefficients from terms in the inverse expansion of the function f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t evaluated at t = 0 for even values of the operation, using a method of Dominici's (nested derivatives, referenced below).
Without proof, appears to be related to computing the 'critical t-value' of Student's t-distribution. (conj.) Critical t-value t_(v, beta) is equal to: sqrt((v/(1-S^2)) - v) where S = (1/2)*Sum_{k>=1} (D^(2*k-2)[f](0)*(1/(2*k-1)!)*(B(1/2, v/2)*(1-2*beta))^(2*k-1)); where (1 - beta) is the confidence interval 'atta' (for a one-tailed distribution such that 'cumulative probability' = t_atta, where beta = 1-atta), x = 1 - (v/2), v: degrees of freedom, B(1/2, v/2) = gamma(1/2)*gamma(v/2)/gamma(1/2 + v/2), D^(2*k - 2)[f](0) is a polynomial function of 'x' whose coefficients are the terms of this sequence as computed using a method of Dominici's on f(t,x) with respect to t (referenced below).

			Given D^k[f]_(b) = (d/dt [f(t)*D^(k-1)[f](t)])_t = b where D^0[f](b) = 1, then for f(t,x) = (1/2)*(1 - t^2)^(-x) where f(0) = 1/2 one obtains: D^2[f]_(0) = -x/2, D^4[f]_(0) = (x/4)*(7*x - 3), D^6[f]_(0) = -(x/8)*(127*x^2 - 123*x + 30), etc., where b is an arbitrary constant.
Triangle begins:
           1;
           7,          3;
         127,        123,          30;
        4369,       6822,        3579,        630;
      243649,     532542,      439899,     162630,      22680;
    20036983,   56717781,    64697499,   37155267,   10735470,   1247400;
  2280356863, 7959325221, 11656842609, 9165745647, 4079027880, 973580580, 97297200;
         ...

Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, International Journal of Mathematics and Mathematical Sciences, Volume 2003, Issue 58, Pages 3699-3715.
Wikipedia, Student's t-distribution

Cf. A002067, A007019, A025035, A002105.

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)Sum_{k=0..n} A241171(n, k)x^k.

Original entry on oeis.org

1, 1, 8, 154, 5552, 321616, 27325088, 3200979664, 494474723072, 97390246272256, 23820397371219968, 7083386168647642624, 2516691244849530785792, 1052914814802404260765696, 512347915163742179541659648, 286902390859642414913802102784, 183187476890368376930869730803712

Offset: 0

Peter Luschny, Sep 03 2022

nonn

Other special values of this Euler type polynomials are: P(n, -1) = A000364(n); P(n, -1/2) = A002105(n); P(n, 1) = A094088(n), where we always make the assumption that the offset of the sequences is 0. A partition refinement of Joffe's triangle A241171 is A327022.

Cf. A241171, A327022, A000364, A002105, A094088, A269941.

a := n -> 2^n*add(A241171(n, k)*(1/2)^k, k = 0..n):
seq(a(n), n = 0..16);

# Using function PtransMatrix from A269941.
def E(n, v):
    eulr = lambda n: 1 / ((2 * n - 1) * (2 * n))
    norm = lambda n, k: (1 / v)^n * factorial(2 * n)
    P = PtransMatrix(n, eulr, norm)
    return [(-1)^j * sum([v^k * P[j][k] for k in range(j + 1)]) for j in range(n)]
A356900List = lambda n: E(n, -1/2); print(A356900List(17))
# A002105List = lambda n: E(n, 1/2) returns the reduced tangent numbers A002105.

cp's OEIS Frontend

A218221 G.f.: 1/(1-x/(1-4*x/(1-10*x/(1-20*x/(1-35*x/(1-56*x/(1-...))))))), a continued fraction.

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A285484 G.f.: 1/(1 + x/(1 + x^3/(1 + x^6/(1 + x^10/(1 + x^15/(1 + ... + x^(k*(k+1)/2)/(1 + ...))))))), a continued fraction.

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A365674 Triangle read by rows. T(n, k) = ((n - k + 1)*(n - k + 2)/2) * T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

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A094346 Another version of triangular array in A036970: triangle T(n,k), 0<=k<=n, read by rows; given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ...] where DELTA is the operator defined in A084938.

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A096078 Triangle read by rows: T(n,k) = (k+1)*T(n-1,k) + (n-k+1)*T(n,k-1).

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A131638 Increasing binary trees having exactly two vertices with outdegree 1.

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Extensions

A320842 Regular triangle whose rows are the coefficients of the Dominici expansion of f(t,x) = (1/2)*(1 - t^2)^(-x) with respect to t.

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A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)*Sum_{k=0..n} A241171(n, k)*x^k.

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A218221 G.f.: 1/(1-x/(1-4x/(1-10x/(1-20x/(1-35x/(1-56*x/(1-...))))))), a continued fraction.

A365674 Triangle read by rows. T(n, k) = ((n - k + 1)(n - k + 2)/2) T(n, k - 1) + T(n - 1, k) for 0 < k < n, T(n, 0) = 1 and T(n, n) = T(n, n - 1) for n > 0.

A096078 Triangle read by rows: T(n,k) = (k+1)T(n-1,k) + (n-k+1)T(n,k-1).

A356900 a(n) = P(n, 1/2) where P(n, x) = x^(-n)Sum_{k=0..n} A241171(n, k)x^k.