A154343 -id:A154343 - cp's OEIS frontend

A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0, 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0, 0, 79360, 0, -32640, 0, 4032, 0, -240, 0, 10, 0

Offset: 0

Peter Luschny, Jul 09 2009

Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
Equivalently, let the polynomials V_n(x) (n>=0) defined by V_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; the sequence L(k) = -1 - H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k > 0 and L(0)=0; H(k) = 1 if k mod 4 <> 0, otherwise 0.
(1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585.
(2) V_n(1) = 1 - Euler(n).
(3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.
(4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.
(5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.
The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.
The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and
P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.
V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,
W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.
B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,
E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.
The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.
Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - Peter Bala, Mar 07 2024

			Triangle begins:
  [0]    0;
  [1]    1,     0;
  [2]    0,     2,     0;
  [3]   -2,     0,     3,   0;
  [4]    0,    -8,     0,   4,    0;
  [5]   16,     0,   -20,   0,    5,    0;
  [6]    0,    96,     0, -40,    0,    6,    0;
  [7] -272,     0,   336,   0,  -70,    0,    7,  0;
  [8]    0, -2176,     0, 896,    0, -112,    0,  8,  0;
  [9] 7936,     0, -9792,   0, 2016,    0, -168,  0,  9,  0;

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.
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.
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.

V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832,
V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536.
Cf. A153641, A154341, A154342, A154343, A154344, A154345.

# Polynomials V_n(x):
V := proc(n,x) local k,pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k); add(binomial(n,k)*euler(k)*pow(x+1,n-k),k=0..n) - pow(x,n) end:
# Coefficients a(n):
seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t),t,16),t,n),x,k),k=0..n)),n=0..8);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

R = PolynomialRing(QQ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A162660(n,k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012

T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
E.g.f. exp(x*t)*tanh(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2-2)*(t^3/3!) + ...
V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k).

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).

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).

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A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

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

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

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

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

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