A153641 -id:A153641 - cp's OEIS frontend

A119881 Expansion of e.g.f. exp(3x)sech(x).

1, 3, 8, 18, 32, 48, 128, 528, 512, -6912, 2048, 357888, 8192, -22351872, 32768, 1903822848, 131072, -209865080832, 524288, 29088886161408, 2097152, -4951498048929792, 8388608, 1015423886523629568, 33554432, -246921480190140874752, 134217728, 70251601603944228323328

Offset: 0

Paul Barry, May 26 2006

Transform of 3^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=3. - Peter Luschny, Nov 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A027641, A027642, A119879, A119880, A153641.

EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
A119881:= func< n| (-2)^n*EulerPoly(n,-1) >;
[A119881(n): n in [0..40]]; // G. C. Greubel, Jun 07 2023

a := proc(n) add(binomial(n,k)*bernoulli(k,1)*2^(n+k)/(n-k+1),k=0..n) end: # Peter Luschny, Dec 14 2008
a := n -> 2^n*abs(euler(n,-1)):  # Peter Luschny, Jan 25 2009
P := proc(n,x) option remember; if n = 0 then 1 else
   (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
   expand(%) fi end:
A119881 := n -> subs(x=-1,P(n,x)):
seq(A119881(n), n=0..27);  # Peter Luschny, Mar 07 2014

Table[2^(n+1) (Zeta[-n] (2^(n+1)-1)+1), {n,0,27}] (* Peter Luschny, Jul 16 2013 *)
Range[0, 30]! CoefficientList[Series[Exp[3 x] Sech[x], {x, 0, 30}], x] (* Vincenzo Librandi, Mar 08 2014 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(3*x)/cosh(x))) \\ Joerg Arndt, Apr 20 2013

def skp(n, x):
    A = lambda k: 0 if (k+1)%4 == 0 else (-1)^((k+1)//4)*2^(-(k//2))
    return add(A(k)*add((-1)^v*binomial(k,v)*(v+x+1)^n for v in (0..k)) for k in (0..n))
A119881 = lambda n: skp(n,3)
[A119881(n) for n in (0..27)]  # Peter Luschny, Nov 23 2012

a(n) = Sum_{k=0..n} A119879(n,k)*3^k.
a(n) = Sum_{k=0..n} binomial(n,k)*B(k,1)*2^(n+k)/(n-k+1). Here B(k,1) are the Bernoulli number A027641(k)/A027642(k) with the exception B(1,1)=1/2. - Peter Luschny, Dec 14 2008
a(n) = 2^n |E(n,-1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
The odd part of a(n) = numerator(Euler(n,2)/2) = 1, 3, 1, 9, 1, 3, 1, 33, 1, -27, 1, 699, ... (compare A143074). - Peter Luschny, Nov 23 2012
G.f.: 1/Q(0), where Q(k) = 1 - 2*x - x*(k+1)/(1+x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - 4*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
a(n) = 2^(n+1)*(zeta[-n]*(2^(n+1)-1)+1). - Peter Luschny, Jul 16 2013
E.g.f.: 2/Q(0), where Q(k) = 1 + 2^k/( 1 - 2*x/( 2*x - 2^k*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
a(n) = 2^(n+1)*(1+(-1)^n*(2^(n+1)-1)*Bernoulli(n+1)/(n+1)). - Vladimir Reshetnikov, Oct 21 2015

A154341 E(n,k), an additive decomposition of the Euler number (triangle read by rows).

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -7, 6, 0, 1, -15, 25, 0, -6, 1, -31, 90, 0, -90, 30, 1, -63, 301, 0, -840, 630, -90, 1, -127, 966, 0, -6300, 7980, -2520, 0, 1, -255, 3025, 0, -41706, 79380, -41580, 0, 2520

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=0 these polynomials result in a decomposition of the Euler number A122045.

			Triangle begins:
  1,
  1,   -1,
  1,   -3,    1,
  1,   -7,    6, 0,
  1,  -15,   25, 0,     -6,
  1,  -31,   90, 0,    -90,    30,
  1,  -63,  301, 0,   -840,   630,    -90,
  1, -127,  966, 0,  -6300,  7980,  -2520,  0,
  1, -255, 3025, 0, -41706, 79380, -41580,  0, 2520,
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows
Peter Luschny, The Swiss-Knife polynomials.

Cf. A122045, A153641, A154342, A154343, A154344, A154345.

E := proc(n,k) local v,c; c := m -> if irem(m+1,4) = 0 then 0 else 1/((-1)^iquo(m+1,4)*2^iquo(m,2)) fi; add((-1)^(v)*binomial(k,v)*c(k)*(v+1)^n,v=0..k) end: seq(print(seq(E(n,k),k=0..n)),n=0..8);

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; e[n_, k_] := Sum[(-1)^v*Binomial[k, v]*c[k]*(v+1)^n, {v, 0, k}]; Table[e[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
E(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*c(k)*(v+1)^n,
A122045(n) = Sum_{k=0..n} E(n,k).

A154342 T(n,k) an additive decomposition of the signed tangent number (triangle read by rows).

Original entry on oeis.org

1, 2, -1, 4, -5, 1, 8, -19, 9, 0, 16, -65, 55, 0, -6, 32, -211, 285, 0, -120, 30, 64, -665, 1351, 0, -1470, 810, -90, 128, -2059, 6069, 0, -14280, 13020, -3150, 0

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1 these polynomials result in a decomposition of the signed tangent numbers A009006.

			Triangle begins:
    1,
    2,    -1,
    4,    -5,    1,
    8,   -19,    9, 0,
   16,   -65,   55, 0,     -6,
   32,  -211,  285, 0,   -120,    30,
   64,  -665, 1351, 0,  -1470,   810,   -90,
  128, -2059, 6069, 0, -14280, 13020, -3150, 0,
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows
Peter Luschny, The Swiss-Knife polynomials.

Cf. A009006, A155585.

Cf. A153641, A154341, A154343, A154344, A154345.

T := proc(n,k) local v,c; c := m -> if irem(m+1,4) = 0 then 0 else 1/((-1)^iquo(m+1,4)*2^iquo(m,2)) fi; add((-1)^(v)*binomial(k,v)*c(k)*(v+2)^n,v=0..k) end: seq(print(seq(T(n,k),k=0..n)),n=0..8);

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; t[n_, k_] := Sum[(-1)^v*Binomial[k, v]*c[k]*(v+2)^n, {v, 0, k}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
T(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*c(k)*(v+2)^n.
A155585(n) = Sum_{k=0..n} T(n,k).

A154343 S(n,k) an additive decomposition of the Springer number (generalized Euler number), (triangle read by rows).

Original entry on oeis.org

1, 3, -2, 9, -16, 4, 27, -98, 60, 0, 81, -544, 616, 0, -96, 243, -2882, 5400, 0, -3360, 960, 729, -14896, 43564, 0, -72480, 46080, -5760, 2187, -75938, 334740, 0, -1246560, 1323840, -362880, 0, 6561, -384064, 2495056, 0, -18801216, 29675520

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1/2 and multiplied by 2^n these polynomials result in a decomposition of the Springer numbers A001586.

			Triangle begins:
  1,
  3,    -2,
  9,    -16,     4,
  27,   -98,     60,      0,
  81,   -544,    616,     0, -96,
  243,  -2882,   5400,    0, -3360,     960,
  729,  -14896,  43564,   0, -72480,    46080,    -5760,
  2187, -75938,  334740,  0, -1246560,  1323840,  -362880,   0,
  6561, -384064, 2495056, 0, -18801216, 29675520, -13386240, 0, 645120,
  ...

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Luschny, The Swiss-Knife polynomials.

Cf. A001586, A188458.

Cf. A153641, A154341, A154342, A154344, A154345.

S := proc(n,k) local v,c; c := m -> if irem(m+1,4) = 0 then 0 else 1/((-1)^iquo(m+1,4)*2^iquo(m,2)) fi; add((-1)^(v)*binomial(k,v)*2^n*c(k)*(v+3/2)^n,v=0..k) end: seq(print(seq(S(n,k),k=0..n)),n=0..8);

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; s[n_, k_] := Sum[(-1)^v*Binomial[k, v]*2^n*c[k]*(v+3/2)^n, {v, 0, k}]; Table[s[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
S(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*2^n*c(k)*(v+3/2)^n.
A188458(n) = Sum_{k=0..n} S(n,k).

A154344 Triangle read by rows. G(n, k) an additive decomposition of 2^n*G(n), G(n) the Genocchi numbers.

Original entry on oeis.org

1, 0, -2, 0, -3, 3, 0, -4, 12, 0, 0, -5, 35, 0, -30, 0, -6, 90, 0, -360, 180, 0, -7, 217, 0, -2730, 3150, -630, 0, -8, 504, 0, -16800, 33600, -15120, 0, 0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680, 0, -10, 2550, 0, -466200, 2085300, -2381400, 0, 907200, -226800

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=-1 multiplied by n+1 this results in a decomposition of 2^n times the Genocchi numbers A036968.

			Triangle begins:
  1,
  0, -2,
  0, -3,    3,
  0, -4,   12, 0,
  0, -5,   35, 0,    -30,
  0, -6,   90, 0,   -360,    180,
  0, -7,  217, 0,  -2730,   3150,    -630,
  0, -8,  504, 0, -16800,  33600,  -15120, 0,
  0, -9, 1143, 0, -91854, 283500, -215460, 0, 22680,
  ...

G. C. Greubel, Table of n, a(n) for n = 0..1274
Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Peter Luschny, The Swiss-Knife polynomials.

Cf. A036968, A153641, A154341, A154342, A154343, A154345.

G := proc(n, k) local v, c; c := m -> if irem(m+1, 4) = 0 then 0 else 1/((-1)^iquo(m+1, 4)*2^iquo(m, 2)) fi; add((-1)^v*binomial(k, v)*(n+1)*c(k)*v^n, v=0..k) end: seq(print(seq(G(n, k), k=0..n)), n=0..8);

g[n_, k_] := Module[{v, c, pow}, pow[a_, b_] := If[ a == 0 && b == 0, 1, a^b]; c[m_] := If[ Mod[m+1, 4] == 0 , 0 , 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; Sum[(-1)^v*Binomial[k, v]*(n+1)*c[k]*pow[v, n], {v, 0, k}]]; Table[g[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2013, translated from Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
G(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*(n+1)*c(k)*v^n.
A036968(n) = (1/2^n)*Sum_{k=0..n} G(n,k).

A154345 B(n,k) an additive decomposition of (4^n-2^n)*B(n), B(n) the Bernoulli numbers (triangle read by rows).

Original entry on oeis.org

1, 4, -2, 12, -15, 3, 32, -76, 36, 0, 80, -325, 275, 0, -30, 192, -1266, 1710, 0, -720, 180, 448, -4655, 9457, 0, -10290, 5670, -630, 1024, -16472, 48552, 0, -114240, 104160, -25200, 0

Offset: 0

Peter Luschny, Jan 07 2009

The Swiss-Knife polynomials A153641 can be understood as a sum of polynomials. Evaluated at x=1 and multiplied by n this results in a decomposition of (4^n-2^n) times the Bernoulli numbers A027641/A027642 (for n>0 and B_1 = 1/2).

			Triangle begins:
  1,
  4,    -2,
  12,   -15,    3,
  32,   -76,    36,    0,
  80,   -325,   275,   0, -30,
  192,  -1266,  1710,  0, -720,    180,
  448,  -4655,  9457,  0, -10290,  5670,   -630,
  1024, -16472, 48552, 0, -114240, 104160, -25200, 0,
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows
Peter Luschny, The Swiss-Knife polynomials.

Cf. A153641, A154341, A154342, A154343, A154344.

B := proc(n,k) local v,c; c := m -> if irem(m+1,4) = 0 then 0 else 1/((-1)^iquo(m+1,4)*2^iquo(m,2)) fi; add((-1)^(v)*binomial(k,v)*n*c(k)*(v+2)^(n-1),v=0..k) end: seq(print(seq(B(n,k),k=0..(n-1))),n=0..8);

c[m_] := If[Mod[m+1, 4] == 0, 0, 1/((-1)^Quotient[m+1, 4]*2^Quotient[m, 2])]; b[n_, k_] := Sum[(-1)^v*Binomial[k, v]*n*c[k]*(v+2)^(n-1), {v, 0, k}]; Table[b[n, k], {n, 0, 8}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jul 30 2013, after Maple *)

Let c(k) = ((-1)^floor(k/4) / 2^floor(k/2)) * [4 not div k+1] (Iverson notation).
B(n,k) = Sum_{v=0..k} (-1)^v*binomial(k,v)*n*c(k)*(v+2)^(n-1).
B(n) = (Sum_{k=0..n} B(n,k)) / (4^n-2^n).

A162590 Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 0, 4, 0, 4, 0, 1, 0, 10, 0, 5, 0, 0, 6, 0, 20, 0, 6, 0, 1, 0, 21, 0, 35, 0, 7, 0, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 36, 0, 126, 0, 84, 0, 9, 0, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 1, 0, 55, 0, 330, 0, 462, 0, 165, 0, 11, 0, 0, 12, 0, 220, 0, 792, 0, 792, 0

Offset: 0

Peter Luschny, Jul 07 2009

Comment from Peter Bala (Dec 06 2011): "Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for" A196776(n,k), which gives the number of ordered partitions of an n set into k odd-sized blocks. - Peter Luschny, Dec 06 2011

The n-th row of the triangle is formed by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probability of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms in the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k) as the (2k)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k). - Luca Onnis, Oct 29 2023

			Triangle begins:
  0
  1,  0
  0,  2,  0
  1,  0,  3,  0
  0,  4,  0,  4,  0
  1,  0, 10,  0,  5,  0
  0,  6,  0, 20,  0,  6,  0
  1,  0, 21,  0, 35,  0,  7,  0
  ...
  p[0](x) = 0;
  p[1](x) = 1
  p[2](x) = 2*x
  p[3](x) = 3*x^2 +  1
  p[4](x) = 4*x^3 +  4*x
  p[5](x) = 5*x^4 + 10*x^2 +  1
  p[6](x) = 6*x^5 + 20*x^3 +  6*x
  p[7](x) = 7*x^6 + 35*x^4 + 21*x^2 + 1
  p[8](x) = 8*x^7 + 56*x^5 + 56*x^3 + 8*x
.
Cf. the triangle of odd-numbered terms in rows of Pascal's triangle (A034867).
p[n] (k), n=0,1,...
k=0:  0, 1,  0,   1,    0,     1, ... A000035, (A059841)
k=1:  0, 1,  2,   4,    8,    16, ... A131577, (A000079)
k=2:  0, 1,  4,  13,   40,   121, ... A003462
k=3:  0, 1,  6,  28,  120,   496, ... A006516
k=4:  0, 1,  8,  49,  272,  1441, ... A005059
k=5:  0, 1, 10,  76,  520,  3376, ... A081199, (A016149)
k=6:  0, 1, 12, 109,  888,  6841, ... A081200, (A016161)
k=7:  0, 1, 14, 148, 1400, 12496, ... A081201, (A016170)
k=8:  0, 1, 16, 193, 2080, 21121, ... A081202, (A016178)
k=9:  0, 1, 18, 244, 2952, 33616, ... A081203, (A016186)
k=10: 0, 1, 20, 301, 4040, 51001, ... ......., (A016190)
.
p[n] (k), k=0,1,...
p[0]: 0,  0,   0,    0,    0,     0, ... A000004
p[1]: 1,  1,   1,    1,    1,     1, ... A000012
p[2]: 0,  2,   4,    6,    8,    10, ... A005843
p[3]: 1,  4,  13,   28,   49,    76, ... A056107
p[4]: 0,  8,  40,  120,  272,   520, ... A105374
p[5]: 1, 16, 121,  496, 1441,  3376, ...
p[6]: 0, 32, 364, 2016, 7448, 21280, ...

Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
Luca Onnis, Animation of the Ehrenfest model.
Wikipedia, Ehrenfest model.

Cf. A119467.

# Polynomials: p_n(x)
p := proc(n,x) local k;
pow := (n,k) -> `if`(n=0 and k=0,1,n^k);
add((k mod 2)*binomial(n,k)*pow(x,n-k),k=0..n) end;
# Coefficients: a(n)
seq(print(seq(coeff(i!*coeff(series(exp(x*t)/csch(t), t,16),t,i),x,n), n=0..i)), i=0..8);

p[n_, x_] := Sum[Binomial[n, 2*k-1]*x^(n-2*k+1), {k, 0, n+2}]; row[n_] := CoefficientList[p[n, x], x] // Append[#, 0]&; Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
n = 15; "n-th row"
mat = Table[Table[0, {j, 1, n + 1}], {i, 1, n + 1}];
mat[[1, 2]] = 1;
mat[[n + 1, n]] = 1;
For[i = 2, i <= n, i++, mat[[i, i - 1]] = (i - 1)/n ];
For[i = 2, i <= n, i++, mat[[i, i + 1]] = (n - i + 1)/n];
mat // MatrixForm;
P2 = Dot[mat, mat];
R1 = Simplify[
  Eigenvectors[Transpose[P2]][[1]]/
   Total[Eigenvectors[Transpose[P2]][[1]]]]
R2 = Table[Dot[R1, Transpose[mat][[k]]], {k, 1, n + 1}]
even = R1*2^(n - 1) (* Luca Onnis, Oct 29 2023 *)

p_n(x) = Sum_{k=0..n} (k mod 2)*binomial(n,k)*x^(n-k).
E.g.f.: exp(x*t)/csch(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2+1)*(t^3/3!) + ...
The 'co'-polynomials with generating function exp(x*t)*sech(t) are the Swiss-Knife polynomials (A153641).

A062393 a(n) = n^5 - (n-1)^5 + (n-2)^5 - ... +(-1)^n*0^5.

Original entry on oeis.org

0, 1, 31, 212, 812, 2313, 5463, 11344, 21424, 37625, 62375, 98676, 150156, 221137, 316687, 442688, 605888, 813969, 1075599, 1400500, 1799500, 2284601, 2869031, 3567312, 4395312, 5370313, 6511063, 7837844, 9372524, 11138625, 13161375

Offset: 0

Henry Bottomley, Jun 21 2001

The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus a(k) = |2^(-6)(P(5,1)-(-1)^k P(5,2k+1))|. - Peter Luschny, Jul 12 2009

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A000539, A000584. A062392 for 4th powers, A152725 for 6th powers.

a := n -> (1-(-1)^n+n^2*(n^2*(2*n+5)-5))/4; # Peter Luschny, Jul 12 2009

k=0;lst={k};Do[k=n^5-k;AppendTo[lst, k], {n, 1, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *)
Table[Total[(Times@@@Partition[Riffle[Range[n,0,-1],{1,-1},{2,-1,2}],2])^5],{n,0,30}] (* or *) LinearRecurrence[ {5,-9,5,5,-9,5,-1},{0,1,31,212,812,2313,5463},40] (* Harvey P. Dale, Feb 01 2013 *)

{ a=0; for (n=0, 1000, write("b062393.txt", n, " ", a=n^5 - a) ) } \\ Harry J. Smith, Aug 07 2009

a(n) = (2*n^5 + 5*n^4 - 5*n^2 + 1 - (-1)^n)/4 = n^5 - a(n-1).
G.f.: x*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1)/((x-1)^6*(x+1)). - Colin Barker, Sep 19 2012
a(0)=0, a(1)=1, a(2)=31, a(3)=212, a(4)=812, a(5)=2313, a(6)=5463, a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7). - Harvey P. Dale, Feb 01 2013

A000810 Expansion of e.g.f. (sin x + cos x)/cos 3x.

Original entry on oeis.org

1, 1, 8, 26, 352, 1936, 38528, 297296, 7869952, 78098176, 2583554048, 31336418816, 1243925143552, 17831101321216, 825787662368768, 13658417358350336, 722906928498737152, 13551022195053101056

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..200

Cf. A007286, A007289.

(-1)^(n*(n-1)/2)*a(n) gives the alternating row sums of A225118. - Wolfdieter Lang, Jul 12 2017

CoefficientList[Series[(Sin[x]+Cos[x])/Cos[3*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)
Table[Abs[EulerE[n, 1/3]] 6^n, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 21 2015 *)

x='x+O('x^66); v=Vec(serlaplace( (sin(x)+cos(x)) / cos(3*x) ) ) \\ Joerg Arndt, Apr 27 2013

from mpmath import mp, lerchphi
mp.dps = 32; mp.pretty = True
def A000810(n): return abs(3^n*2^(n+1)*lerchphi(-1,-n,1/3))
[int(A000810(n)) for n in (0..17)]  # Peter Luschny, Apr 27 2013

a(2n) = A000436(n).
(-1)^n*a(2n+1)=1-sum_{i=0,1,...,n-1} (-1)^i*binomial(2n+1,2i+1)*3^(2n-2i)*a(2i+1). - R. J. Mathar, Nov 19 2006
a(n) = | 3^n*2^(n+1)*lerchphi(-1,-n,1/3) |. - Peter Luschny, Apr 27 2013
a(n) ~ n!*2^(n+1)*3^(n+1/2)/Pi^(n+1) if n is even and a(n) ~ n!*2^(n+1)*3^n/Pi^(n+1) if n is odd. - Vaclav Kotesovec, Jun 25 2013
a(n) = (-1)^floor(n/2)*3^n*skp(n, 1/3), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.

Original entry on oeis.org

0, 2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786

Offset: 0

Peter Bala, Jun 18 2009

Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * Sum_{r = 0..m-1} X(r)*exp(r*t) = Sum_{n >= 0} B_n(X,x)*t^n/n!.
The values B_n(X,0) are generalizations of the Bernoulli numbers (case X = 1). For the theory and properties of these polynomials and numbers see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
The odd-indexed generalized Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even-indexed values B_(2*n)(X,0).
The coefficients of the generalized Bernoulli polynomials B_n(X,x) are listed in A151751.

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

Cf. A000464, A153641, A151751, A117442.

G := x*sinh(x)/cosh(2*x): ser := series(G, x, 30):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..14); # Peter Luschny, Nov 26 2020
# After an observation of F. Chapoton in A117442:
A161722 := proc(n) 4^n*add(binomial(2*n, k)*euler(k)*((x+1)/2)^(2*n-k), k=0..2*n);
coeff(%, x, 1) end: seq(A161722(n), n=0..13); # Peter Luschny, Nov 26 2020

terms = 13;
(CoefficientList[x(Sinh[x]/Cosh[2x]) + O[x]^(2terms+3), x] Range[0, 2terms+2]!)[[ ;; ;; 2]] (* Jean-François Alcover, Nov 16 2020 *)

(1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).
The sequence of generalized Bernoulli numbers
(2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....
Hence
(5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.
a(n) = (-1/2)*16^n*n*euler(2*n-1, 1/4) for n >= 1 after a formula of Peter Bala in A000464. - Peter Luschny, Nov 26 2020

Cross-reference corrected by Peter Bala, Jun 22 2009

Offset set to 0 and a(0) = 0 prepended by Peter Luschny, Nov 26 2020

cp's OEIS Frontend

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