A121670 -id:A121670 - cp's OEIS frontend

A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

1, 1, 1, -1, 1, -3, 1, -6, 5, 1, -10, 25, 1, -15, 75, -61, 1, -21, 175, -427, 1, -28, 350, -1708, 1385, 1, -36, 630, -5124, 12465, 1, -45, 1050, -12810, 62325, -50521, 1, -55, 1650, -28182, 228525, -555731, 1, -66, 2475, -56364, 685575, -3334386, 2702765, 1

Offset: 0

Peter Luschny, Dec 29 2008

In the following the expression [n odd] is 1 if n is odd, 0 otherwise.
(+) W_n(0) = E_n are the Euler (or secant) numbers A122045.
(+) W_n(1) = T_n are the signed tangent numbers, see A009006.
(+) W_{n-1}(1) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli number A027641/A027642.
(+) W_n(-1) 2^{-n}(n+1) = G_n the Genocchi number A036968.
(+) W_n(1/2) 2^{n} are the signed generalized Euler (Springer) number, see A001586.
(+) | W_n([n odd]) | the number of alternating permutations A000111.
(+) | W_n([n odd]) / n! | for 0<=n the Euler zeta number A099612/A099617 (see Wikipedia on Bernoulli number). - Peter Luschny, Dec 29 2008
The diagonals in the full triangle (with zero coefficients) of the polynomials have the general form E(k)*binomial(n+k,k) (k>=0 fixed, n=0,1,...) where E(n) are the Euler numbers in the enumeration A122045. For k=2 we find the triangular numbers A000217 and for k=4 A154286. - Peter Luschny, Jan 06 2009
From Peter Bala, Jun 10 2009: (Start)
The Swiss-Knife polynomials W_n(x) may be expressed in terms of the Bernoulli polynomials B(n,x) as
... W_n(x) = 4^(n+1)/(2*n+2)*[B(n+1,(x+3)/4) - B(n+1,(x+1)/4)].
The Swiss-Knife polynomials are, apart from a multiplying factor, examples of generalized Bernoulli polynomials.
Let X be the Dirichlet character modulus 4 defined by X(4*n+1) = 1, X(4*n+3) = -1 and X(2*n) = 0. The generalized Bernoulli polynomials B(X;n,x), n = 1,2,..., associated with the character X are defined by means of the generating function
... t*exp(x*t)*(exp(t)-exp(3*t))/(exp(4*t)-1) = sum {n = 1..inf} B(X;n,x)*t^n/n!.
The first few values are B(X;1,x) = -1/2, B(X;2,x) = -x, B(X,3,x) = -3/2*(x^2-1) and B(X;4,x) = -2*(x^3-3*x).
In general, W_n(x) = -2/(n+1)*B(X;n+1,x).
For the theory of generalized Bernoulli polynomials associated to a periodic arithmetical function see [Cohen, Section 9.4].
The generalized Bernoulli polynomials may be used to evaluate twisted sums of k-th powers. For the present case the result is
sum{n = 0..4*N-1} X(n)*n^k = 1^k - 3^k + 5^k - 7^k + ... - (4*N-1)^k
= [B(X;k+1,4*N) - B(X;k+1,0)]/(k+1) = [W_k(0) - W_k(4*N)]/2.
For the proof apply [Cohen, Corollary 9.4.17 with m = 4 and x = 0].
The generalized Bernoulli polynomials and the Swiss-Knife polynomials are also related to infinite sums of powers through their Fourier series - see the formula section below. For a table of the coefficients of generalized Bernoulli polynomials attached to a Dirichlet character modulus 8 see A151751.
(End)
The Swiss-Knife polynomials provide a general formula for alternating sums of powers similar to the formula which are provided by the Bernoulli polynomials for non-alternating sums of powers (see the Luschny link). Sequences covered by this formula include A001057, A062393, A062392, A011934, A144129, A077221, A137501, A046092. - Peter Luschny, Jul 12 2009
The greatest common divisor of the nonzero coefficients of the decapitated Swiss-Knife polynomials is exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd primes, symbolically:
gcd(coeffs(SKP_{n}(x) - x^n)) = A155457(n) (n>1). - Peter Luschny, Dec 16 2009
Another version is at A119879. - Philippe Deléham, Oct 26 2013

			1
x
x^2  -1
x^3  -3x
x^4  -6x^2   +5
x^5 -10x^3  +25x
x^6 -15x^4  +75x^2  -61
x^7 -21x^5 +175x^3 -427x

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. [From Peter Bala, Jun 10 2009]

G. C. Greubel, Table of n, a(n) for the first 76 rows, flattened
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.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A Oct. 2007 vol 463 no. 463 2086 2401-2414. [Added by Tom Copeland, Aug 31 2015]
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife polynomials and Euler numbers
Wikipedia, Bernoulli number
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

Cf. A151751, A162590, A000111, A001586, A009006, A027641/A027642, A036968, A099612/A099617, A119879, A104033.
W_n(k), k=0,1,...
W_0:  1,  1,  1,  1,   1,   1, ........ A000012
W_1:  0,  1,  2,  3,   4,   5, ........ A001477
W_2: -1,  0,  3,  8,  15,  24, ........ A067998
W_3:  0, -2,  2, 18,  52, 110, ........ A121670
W_4:  5,  0, -3, 32, 165, 480, ........
W_n(k), n=0,1,...
k=0:  1,  0, -1,  0,   5,   0, -61, ... A122045
k=1:  1,  1,  0, -2,   0,  16,   0, ... A155585
k=2:  1,  2,  3,  2,  -3,   2,  63, ... A119880
k=3:  1,  3,  8, 18,  32,  48, 128, ... A119881
k=4:  1,  4, 15, 52, 165, 484, ........         [Peter Luschny, Jul 07 2009]

w := proc(n,x) local v,k,pow,chen; pow := (a,b) -> if a = 0 and b = 0 then 1 else a^b fi; chen := proc(m) if irem(m+1,4) = 0 then RETURN(0) fi; 1/((-1)^iquo(m+1,4) *2^iquo(m,2)) end; add(add((-1)^v*binomial(k,v)*pow(v+x+1,n)*chen(k),v=0..k), k=0..n) end:
# Coefficients with zeros:
seq(print(seq(coeff(i!*coeff(series(exp(x*t)*sech(t),t,16),t,i),x,i-n),n=0..i)), i=0..8);
# Recursion
W := proc(n,z) option remember; local k,p;
if n = 0 then 1 else p := irem(n+1,2);
z^n - p + add(`if`(irem(k,2)=1,0,
W(k,0)*binomial(n,k)*(power(z,n-k)-p)),k=2..n-1) fi end:
# Peter Luschny, edited and additions Jul 07 2009, May 13 2010, Oct 24 2011

max = 9; rows = (Reverse[ CoefficientList[ #, x]] & ) /@ CoefficientList[ Series[ Exp[x*t]*Sech[t], {t, 0, max}], t]*Range[0, max]!; par[coefs_] := (p = Partition[ coefs, 2][[All, 1]]; If[ EvenQ[ Length[ coefs]], p, Append[ p, Last[ coefs]]]); Flatten[ par /@ rows] (* Jean-François Alcover, Oct 03 2011, after g.f. *)
sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; Table[CoefficientList[sk[n, x], x] // Reverse // Select[#, # =!= 0 &] &, {n, 0, 13}] // Flatten (* Jean-François Alcover, May 21 2013 *)
Flatten@Table[Binomial[n, 2k] EulerE[2k], {n, 0, 12}, {k, 0, n/2}](* Oliver Seipel, Jan 14 2025 *)

def A046978(k):
    if k % 4 == 0:
        return 0
    return (-1)**(k // 4)
def A153641_poly(n, x):
    return expand(add(2**(-(k // 2))*A046978(k+1)*add((-1)**v*binomial(k,v)*(v+x+1)**n for v in (0..k)) for k in (0..n)))
for n in (0..7): print(A153641_poly(n, x))  # Peter Luschny, Oct 24 2011

W_n(x) = Sum_{k=0..n}{v=0..k} (-1)^v binomial(k,v)*c_k*(x+v+1)^n where c_k = frac((-1)^(floor(k/4))/2^(floor(k/2))) [4 not div k] (Iverson notation).
From Peter Bala, Jun 10 2009: (Start)
E.g.f.: 2*exp(x*t)*(exp(t)-exp(3*t))/(1-exp(4*t))= 1 + x*t + (x^2-1)*t^2/2! + (x^3-3*x)*t^3/3! + ....
W_n(x) = 1/(2*n+2)*Sum_{k=0..n+1} 1/(k+1)*Sum_{i=0..k} (-1)^i*binomial(k,i)*((x+4*i+3)^(n+1) - (x+4*i+1)^(n+1)).
Fourier series expansion for the generalized Bernoulli polynomials:
B(X;2*n,x) = (-1)^n*(2/Pi)^(2*n)*(2*n)! * {sin(Pi*x/2)/1^(2*n) - sin(3*Pi*x/2)/3^(2*n) + sin(5*Pi*x/2)/5^(2*n) - ...}, valid for 0 <= x <= 1 when n >= 1.
B(X;2*n+1,x) = (-1)^(n+1)*(2/Pi)^(2*n+1)*(2*n+1)! * {cos(Pi*x/2)/1^(2*n+1) - cos(3*Pi*x/2)/3^(2*n+1) + cos(5*Pi*x/2)/5^(2*n+1) - ...}, valid for 0 <= x <= 1 when n >= 1 and for 0 <= x < 1 when n = 0.
(End)
E.g.f.: exp(x*t) * sech(t). - Peter Luschny, Jul 07 2009
O.g.f. as a J-fraction: z/(1-x*z+z^2/(1-x*z+4*z^2/(1-x*z+9*z^2/(1-x*z+...)))) = z + x*z^2 + (x^2-1)*z^3 + (x^3-3*x)*z^4 + .... - Peter Bala, Mar 11 2012
Conjectural o.g.f.: Sum_{n >= 0} (1/2^((n-1)/2))*cos((n+1)*Pi/4)*( Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - (k + x)*t) ) = 1 + x*t + (x^2 - 1)*t^2 + (x^3 - 3*x)*t^3 + ... (checked up to O(t^13)), which leads to W_n(x) = Sum_{k = 0..n} 1/2^((k - 1)/2)*cos((k + 1)*Pi/4)*( Sum_{j = 0..k} (-1)^j*binomial(k, j)*(j + x)^n ). - Peter Bala, Oct 03 2016

A080663 a(n) = 3*n^2 - 1.

Original entry on oeis.org

2, 11, 26, 47, 74, 107, 146, 191, 242, 299, 362, 431, 506, 587, 674, 767, 866, 971, 1082, 1199, 1322, 1451, 1586, 1727, 1874, 2027, 2186, 2351, 2522, 2699, 2882, 3071, 3266, 3467, 3674, 3887, 4106, 4331, 4562, 4799, 5042, 5291, 5546, 5807, 6074, 6347, 6626

Offset: 1

Cino Hilliard, Mar 01 2003

These numbers cannot be perfect squares. See the Hilliard link for a proof.
2nd elementary symmetric polynomial of n, n + 1 and n + 2: n(n+1) + n(n+2) + (n+1)(n+2). - Zak Seidov, Mar 23 2005
This sequence equals for n >= 2 the third right hand column of triangle A165674. Its recurrence relation leads to Pascal's triangle A007318. Crowley's formula for A080663(n-1) leads to Wiggen's triangle A028421 and the o.g.f. of this sequence, without the first term, leads to Wood's polynomials A126671. See also A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 16 2009
The Diophantine equation x(x+1) + (x+2)(x+3) = (x+y)^2 + (x-y)^2 has solutions x = a(n), y = 3n. - Bruno Berselli, Mar 29 2013
A simpler proof that these numbers can't be perfect squares can easily be constructed using congruences: If the equation x^2 = 3y^2 - 1 has a solution in positive integers, then x^2 = 2 mod 3. Obviously we can't have x = 0 mod 3, and x = 1 mod 3 doesn't work either because then x^2 = 1 mod 3 also. That leaves x = 2 mod 3, but then x^2 = 1 mod 3. - Alonso del Arte, Oct 19 2013
2*a(n+1) is surface area of a rectangular prism with consecutive integer sides: n, n+1, and n+2, (n>0). - Wesley Ivan Hurt, Sep 06 2014
Numbers m such that 3*m+3 is a square. So, these are the numbers m such that the system of equations x=sqrt(m-2yz), y=sqrt(m+1-2xz), z=sqrt(m+2-2xy) admits 3 real positive solutions whose sum is an integer. See the Rechtman link. - Michel Marcus, Jun 06 2020

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.6.
E. Grosswald, Topics from the Theory of Numbers, 1966 p 64 problem 11

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Cino Hilliard, 3n^2 - 1 not square. [Archived copy as of Apr 11 2008 from web.archive.org]
Ana Rechtman, Juin 2020, 1er défi, Images des Mathématiques, CNRS, 2020 (in French).
Leo Tavares, Illustration: Conjoined Trapezoids
Eric Weisstein's World of Mathematics, Symmetric Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A007318, A028421, A080663, A121670, A126671, A165674, A165676, A165677, A165678, A165679.

Cf. A005449, A115067.

[3*n^2-1 : n in [1..50]]; // Wesley Ivan Hurt, Sep 04 2014

A080663 := proc(n) return 3*n^2-1: end proc: seq(A080663(n), n=1..50); # Nathaniel Johnston, Oct 16 2013

3*Range[47]^2 - 1 (* Alonso del Arte, Oct 19 2013 *)

list(n) = { for(x=1,n, y = 3*x*x-1; print1(y, ", ") ) } \\ edited by Michel Marcus, Feb 01 2020

Vec(x*(2+5*x-x^2)/(1-x)^3+O(x^66)) \\ Joerg Arndt, Sep 06 2014

a(n) = -Re((1 + n*i)^3) where i=sqrt(-1). - Gary W. Adamson, Aug 14 2006
a(n) = 3*n^2 - 1. - Stephen Crowley, Jul 06 2009
a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3). - Johannes W. Meijer, Oct 16 2009
G.f.: x*(2 + 5*x - x^2)/(1-x)^3. - Joerg Arndt, Sep 06 2014
a(n) = a(n-1) + 6*n - 3 for n > 1. - Vincenzo Librandi, Aug 08 2010
E.g.f.: 1 + exp(x)*(3*x^2 + 3*x - 1). - Stefano Spezia, Feb 01 2020
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(3))*cot(Pi/sqrt(3)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(3))*csc(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(3))*sin(sqrt(2/3)*Pi)/sqrt(2). (End)
a(n) = A005449(n) + A115067(n). - Leo Tavares, May 25 2022
a(n) = (n-1)*n + (n-1)*(n+1) + n*(n+1), for n >= 1. See the Zak Seidov comment above. - Wolfdieter Lang, Aug 15 2024

A121672 Real part of (n + i)^5.

Original entry on oeis.org

0, -4, -38, -12, 404, 1900, 5646, 13412, 27688, 51804, 90050, 147796, 231612, 349388, 510454, 725700, 1007696, 1370812, 1831338, 2407604, 3120100, 3991596, 5047262, 6314788, 7824504, 9609500, 11705746, 14152212, 16990988, 20267404, 24030150

Offset: 0

Gary W. Adamson, Aug 14 2006

The companion sequence A121671 uses the operation (1 + n*i)^5.

			a(4) = 404 since (4 + i)^5 = (404 + 1121i) where 1121 = A121671(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A121671, A080663, A121670.

Re[(Range[0,30]+I)^5] (* Harvey P. Dale, Mar 01 2012 *)

a(n) = real((n + I)^5); \\ Michel Marcus, Dec 19 2020

G.f.: -2*x*(2+7*x-78*x^2+7*x^3+2*x^4)/(1-x)^6. - Bruno Berselli, Mar 01 2012
a(n) = n*(n^4-10*n^2+5). - Bruno Berselli, Mar 01 2012
a(n) = (1+n^2)^(5/2)*sin(5*arctan(n)). - Gerry Martens, Apr 06 2024

Corrected and extended by Harvey P. Dale, Mar 01 2012

A244490 Triangle read by rows: T(n,k) (0 <= k <= n) = Sum_{i=0..[k/2]} (-1)^ibinomial(k,2i)(2i-1)!!n^(k-2i).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 18, 1, 4, 15, 52, 163, 1, 5, 24, 110, 478, 1950, 1, 6, 35, 198, 1083, 5706, 28821, 1, 7, 48, 322, 2110, 13482, 83824, 505876, 1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569, 1, 9, 80, 702, 6078, 51894, 436656, 3618540, 29510268, 236644092, 1, 10, 99, 970, 9403, 90150, 854485, 8003950, 74058105, 676549450, 6098971555

Offset: 0

N. J. A. Sloane, Jul 05 2014

East and Gray (p. 24) give a combinatorial interpretation of the numbers: A function f: Y -> X with Y <= X (<= inclusion) has a 2-cycle if there exists x, y in Y with x != y, f(x) = y and f(y) = x. Then T(n,k) = card({f : [k] -> [n]: f has no 2-cycles}). For instance T(3,3) = 18 because there are 27 functions [3] -> [3], 9 of which have a 2-cycle. - Peter Luschny, Oct 05 2016

			Triangle begins:
1
1 1
1 2 3
1 3 8 18
1 4 15 52 163
1 5 24 110 478 1950
1 6 35 198 1083 5706 28821
1 7 48 322 2110 13482 83824 505876
1 8 63 488 3715 27768 203569 1461944 10270569
1 9 80 702 6078 51894 436656 3618540 29510268 236644092
...

J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

As has been noticed by Tom Copeland: T(n,0) = A000012(n), T(n,1) = A001477(n) for n>=1, T(n,2) = A067998(n+1) for n>=3, T(n,3) = A121670(n) for n>=3.

Cf. A276999.

T := (n,k) -> add((-1)^i*binomial(k,2*i)*doublefactorial(2*i-1)*n^(k-2*i), i=0..k/2):
seq(seq(T(n,k), k=0..n), n=0..10); # Peter Luschny, Oct 05 2016

Table[Simplify[2^(-k/2) HermiteH[k, n/Sqrt[2]]], {n, 0, 10}, {k,0,n}] // Flatten (* Peter Luschny, Oct 05 2016 *)

def T(n, k):
    @cached_function
    def h(n, x):
        if n == 0: return 1
        if n == 1: return 2*x
        return 2*(x*h(n-1,x)-(n-1)*h(n-2,x))
    return h(k, n/sqrt(2))/2^(k/2)
for n in range(10):
    print([T(n,k) for k in (0..n)]) # Peter Luschny, Oct 05 2016

From Peter Luschny, Oct 05 2016: (Start)
T(n,k) = 2^(-k/2)*HermiteH(k, n/sqrt(2)).
T(n,k) = 2^((k-1)/2)*n*KummerU((1-k)/2, 3/2, n^2/2) for n>=1.
T(n,k) = n^k*hypergeom([-k/2, (1-k)/2], [], -2/n^2) for n>=1. (End)

A231123 Array T(n,k) read by antidiagonals: T(n,k) = sum(i=0...n, (-1)^(n+i) * C(n+i,2i) * n/(2i+1) * k^(2i+1) ), n>0, k>1.

Original entry on oeis.org

2, 2, 18, 2, 123, 52, 2, 843, 724, 110, 2, 5778, 10084, 2525, 198, 2, 39603, 140452, 57965, 6726, 322, 2, 271443, 1956244, 1330670, 228486, 15127, 488, 2, 1860498, 27246964, 30547445, 7761798, 710647, 30248, 702, 2, 12752043, 379501252, 701260565, 263672646

Offset: 2

Ralf Stephan, Nov 04 2013

The polynomial x^(4n+2) - T(n,k)*x^(2n+1) + 1 is reducible. Example: x^10-123x^5+1=(x^2-3x+1)(x^8+3x^7+8x^6+21x^5+55x^4+21x^3+8x^2+3x+1). It is conjectured that for prime p=2n+1, these are the only values where this holds.

			Array starts
2, 18, 52, 110, 198, 322, 488, 702, 970,...
2, 123, 724, 2525, 6726, 15127, 30248, 55449, 95050,...
2, 843, 10084, 57965, 228486, 710647, 1874888, 4379769, 9313930,...
2, 5778, 140452, 1330670, 7761798, 33385282, 116212808, 345946302,...
2, 39603, 1956244, 30547445, 263672646, 1568397607, 7203319208,...

A. Schinzel, On reducible trinomials III. In: Selecta, Vol. I, European Mathematical Society 2007, pp. 625-626.

Ralf Stephan, On a class of reducible trinomials

T(i,k)=n=2*i+1;sum(m=0,(n-1)/2,(-1)^(m+(n-1)/2)*n*binomial((n+2*m+1)/2-1,2*m)/(2*m+1)*k^(2*m+1))

T(,2) = 2, T(1,n) = A121670(n), T(2,n) = A230586(n).

T(n,k) = sum(i=1..n, (-1)^i * A111125(n,i) * k^(2i+1) ).

A272870 Real part of (n + i)^4.

Original entry on oeis.org

1, -4, -7, 28, 161, 476, 1081, 2108, 3713, 6076, 9401, 13916, 19873, 27548, 37241, 49276, 64001, 81788, 103033, 128156, 157601, 191836, 231353, 276668, 328321, 386876, 452921, 527068, 609953, 702236, 804601, 917756, 1042433, 1179388, 1329401, 1493276

Offset: 0

Colin Barker, May 08 2016

a(1) and a(2) are the only two negative terms in the sequence. Since (n + i)^2 = (n^2 - 1) + 2ni, it follows that (n + i)^4 = (n^2 - 1 + 2ni)^2 = (n^4 - 6n^2 + 1) + (4n^3 - 4n)i, so the real part of (n + i)^4 is n^4 - 6n^2 + 1. Then n^4 + 1 > 6n^2 for all n > 2, ensuring a(n) is positive for all n > 2. - Alonso del Arte, Jun 04 2016

			a(5) = 476 because (5 + i)^4 = 476 + 480*i.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A272871 (imaginary part).

Cf. A121670 ((n+i)^3), A121672 ((n+i)^5).

Table[Re[(n + I)^4], {n, 0, 35}] (* or *)
Table[n^4 - 6 n^2 + 1, {n, 0, 35}] (* or *)
CoefficientList[Series[(1 - 9 x + 23 x^2 + 13 x^3 - 4 x^4)/(1 - x)^5, {x, 0, 35}], x] (* Michael De Vlieger, May 08 2016 *)

PARI
```
a(n) = n^4-6*n^2+1
```
PARI
```
vector(50, n, n--; real((n+I)^4))
```

Vec((1-9*x+23*x^2+13*x^3-4*x^4)/(1-x)^5 + O(x^50))

a(n) = n^4 - 6*n^2 + 1.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (1-9*x+23*x^2+13*x^3-4*x^4) / (1-x)^5.
E.g.f.: (1 - 5*x + x^2 + 6*x^3 + x^4)*exp(x). - Ilya Gutkovskiy, May 08 2016

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, -1, 1, 1, 0, 0, 2, 1, 5, -2, 3, 3, 1, 0, 0, 2, 8, 4, 1, -61, 16, -3, 18, 15, 5, 1, 0, 0, 2, 32, 52, 24, 6, 1, 1385, -272, 63, 48, 165, 110, 35, 7, 1, 0, 0, 2, 128, 484, 480, 198, 48, 8, 1, -50521, 7936, -1383, 528, 1395, 2000, 1085, 322, 63, 9, 1

Offset: 0

Peter Luschny, Dec 14 2014

This two-dimensional array of numbers can be seen as a generalization of the Euler secant and Euler tangent numbers (which are in their compressed and signless form A000364 resp. A000182 or interleaved in A000111). The cases n=0 and n=1 reduce to their expanded and signed forms A122045 and A155585. Moreover the columns are the values of the Swiss-Knife polynomials A153641 evaluated at the nonnegative integers.

Subsequences [3,3,1], [8,4,1], [15,5,1], [24,6,1], [35,7,1], [48,8,1], [63,9,1] found in rows of this entry, as a triangular array, are present in the antidiagonals of Table 5 of the East and Gray reference (A244490), and some subsequences in the rows of Table 5 are found in the antidiagonals of this entry, including [3,2,1] and [1,1]. Equivalently, the first four columns of Table 5 are embedded in this entry viewed as a square array on the table page. An explicit formula with combinatorial interpretations for these numbers is provided in the reference, and others are known for the corresponding columns for the modified Hermite polynomials of A244490. - Tom Copeland, Oct 04 2016

			Square array starts:
  [n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]     [8]
  [0]   1, 0, -1,   0,    5,    0,   -61,      0,   1385, ... A122045
  [1]   1, 1,  0,  -2,    0,   16,     0,   -272,      0, ... A155585
  [2]   1, 2,  3,   2,   -3,    2,    63,      2,  -1383, ... A119880
  [3]   1, 3,  8,  18,   32,   48,   128,    528,    512, ... A119881
  [4]   1, 4, 15,  52,  165,  484,  1395,   4372,  14505, ...
  [5]   1, 5, 24, 110,  480, 2000,  8064,  32240, 130560, ... A225116
  [6]   1, 6, 35, 198, 1085, 5766, 29855, 151878, 766745, ...
  A000012, A001477, A067998, A121670, ...
Triangular array starts:
                1,
              0,  1,
           -1,  1,  1,
          0,  0,  2,  1,
        5, -2,  3,  3,  1,
      0,  0,  2,  8,  4,  1,
  -61, 16, -3, 18, 15,  5,  1.

J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Cf. A122045, A155585, A119880, A119881, A225116, A000364, A000182, A000111, A067998, A121670, A153641, A247501.

Cf. A244490.

# EGF (row)
egf := n -> exp(n*x)*sech(x):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..8)), n=0..6);
# Swiss-Knife polynomial (column)
SKP := proc(n, x) local v, k, A; A := k -> `if`(k mod 4 = 0,0,(-1)^iquo(k,4)); add(2^iquo(-k,2)*A(k+1)*add((-1)^v* binomial(k,v)*(v+x+1)^n,v=0..k), k=0..n); expand(%) end:
seq(print(seq(SKP(k, n), n=0..9)), k=0..6);
# OGF (column)
col := proc(n, len) local T; T := A247501_row(n);
(-1)^(n+1)*add(T[k+1]/(x-1)^(k+1),k=0..n);
seq(coeff(series(%,x,len+1),x,j),j=0..len) end:
seq(print(col(n,8)), n=0..6);

nmax = 10; Clear[row]; row[n_] := row[n] = CoefficientList[Exp[n*x]*Sech[x] + O[x]^(nmax+2), x][[1 ;; nmax+1]]*Range[0, nmax]!;
rows = Table[row[n], {n, 0, nmax}];
T[n_, k_] := rows[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 03 2017 *)

G.f. for column k: the k-th column consists of the values of the k-th Swiss-Knife polynomial skp_{k}(x) evaluated at x = 0,1,2,...

O.g.f. for column k: Sum_{j=0..k} (-1)^(k+1)*A247501(k,j)/(x-1)^(j+1).

cp's OEIS Frontend

A136434 Duplicate of A121670.

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A153641 Nonzero coefficients of the Swiss-Knife polynomials for the computation of Euler, tangent, and Bernoulli numbers (triangle read by rows).

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A080663 a(n) = 3*n^2 - 1.

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A121672 Real part of (n + i)^5.

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A244490 Triangle read by rows: T(n,k) (0 <= k <= n) = Sum_{i=0..[k/2]} (-1)^i*binomial(k,2*i)*(2*i-1)!!*n^(k-2*i).

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A231123 Array T(n,k) read by antidiagonals: T(n,k) = sum(i=0...n, (-1)^(n+i) * C(n+i,2i) * n/(2i+1) * k^(2i+1) ), n>0, k>1.

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A272870 Real part of (n + i)^4.

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A244490 Triangle read by rows: T(n,k) (0 <= k <= n) = Sum_{i=0..[k/2]} (-1)^ibinomial(k,2i)(2i-1)!!n^(k-2i).

A247498 Generalized Euler numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k] exp(nx)*sech(x), n>=0, k>=0.