A001489 -id:A001489 - cp's OEIS frontend

A319932 a(n) = (1/720)n(n - 10)(n - 1)(n^3 - 34n^2 + 181n - 144).

0, 0, -2, -7, -10, -5, 11, 35, 56, 54, 0, -143, -418, -871, -1547, -2485, -3712, -5236, -7038, -9063, -11210, -13321, -15169, -16445, -16744, -15550, -12220, -5967, 4158, 19285, 40745, 70091, 109120, 159896, 224774, 306425, 407862, 532467, 684019, 866723

Offset: 0

Peter Luschny, Oct 02 2018

sign

Cf. A000012 (m=0), A001489 (m=1), A080956 (m=2), A167541 (m=3), A319930 (m=4), A319931 (m=5), this sequence (m=6).

Cf. A319933.

a := n -> (1/720)*n*(n-10)*(n-1)*(n^3-34*n^2+181*n-144);
seq(a(n), n=0..39);

a(n) = [x^5] DedekindEta(x)^n.

a(n) = A319933(n, 5).

A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + kx/(1 + kx^2/(1 + kx^3/(1 + kx^4/(1 + k*x^5/(1 + ...)))))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 4, 0, 0, 1, -4, 9, -4, -1, 0, 1, -5, 16, -18, 0, 1, 0, 1, -6, 25, -48, 27, 8, -1, 0, 1, -7, 36, -100, 128, -27, -24, 1, 0, 1, -8, 49, -180, 375, -320, -27, 48, 0, 0, 1, -9, 64, -294, 864, -1375, 704, 243, -64, -1, 0, 1, -10, 81, -448, 1715, -4104, 4875, -1280, -810, 48, 2, 0

Offset: 0

Ilya Gutkovskiy, May 16 2017

			G.f. of column k: A(x) = 1 - k*x + k^2*x^2 - (k - 1)*k^2*x^3 + (k - 2)*k^3*x^4 - k^3*(k^2 - 3*k + 1)*x^5 + ...
Square array begins:
  1,  1,  1,   1,    1,     1,  ...
  0, -1, -2,  -3,   -4,    -5,  ...
  0,  1,  4,   9,   16,    25,  ...
  0,  0, -4, -18,  -48,  -100,  ...
  0, -1,  0,  27,  128,   375,  ...
  0,  1,  8, -27, -320, -1375,  ...

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Columns k=0..1 give: A000007, A007325.
Rows n=0..3 give: A000012, A001489, A000290, A045991 (gives absolute value).
Main diagonal gives A291335.
Cf. A286509.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))), a continued fraction.

G.f. of column k (for k > 0): (Sum_{j>=0} k^j*x^(j*(j+1))/Product_{i=1..j} (1 - x^i)) / (Sum_{j>=0} k^j*x^(j^2)/Product_{i=1..j} (1 - x^i)).

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset: 0

Peter Luschny, Aug 04 2019

A formal extension of the figurative numbers A139600 to negative n.

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A291701 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...))))).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 1, -1, 0, 1, -4, 3, -2, 0, 0, 1, -5, 6, -4, 2, -1, 0, 1, -6, 10, -8, 6, -2, -1, 0, 1, -7, 15, -15, 13, -6, 1, -1, 0, 1, -8, 21, -26, 25, -16, 6, 0, -2, 0, 1, -9, 28, -42, 45, -36, 18, -3, 0, -2, 0, 1, -10, 36, -64, 77, -72

Offset: 0

Seiichi Manyama, Aug 30 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0,  0,  1,  3,  6, ...
   0, -1, -2, -4, -8, ...
   0,  0,  2,  6, 13, ...

Columns k=0..1 give A000007, A291148.
Rows n=0..1 give A000012, A001489.
Main diagonal gives A291702.
Cf. A291652.

G.f. of column k: (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...))))))^k, a continued fraction.

A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 4, -6, 0, 1, 5, -10, 0, 5, 0, 6, -15, 0, 15, 0, -3, 7, -21, 0, 35, 0, -21, 0, 8, -28, 0, 70, 0, -84, 0, 17, 9, -36, 0, 126, 0, -252, 0, 153, 0, 10, -45, 0, 210, 0, -630, 0, 765, 0, -155, 11, -55, 0, 330, 0, -1386, 0, 2805, 0, -1705, 0, 12, -66, 0, 495

Offset: 1

Ralf Stephan, Sep 08 2004

The Genocchi numbers A001489 appear as constant term of every second polynomial and as the negative sum of its coefficients.

			G(1,x) = 1
G(2,x) = 2*x - 1
G(3,x) = 3*x^2 - 3*x
G(4,x) = 4*x^3 - 6*x^2 + 1
G(5,x) = 5*x^4 - 10*x^3 + 5*x
G(6,x) = 6*x^5 - 15*x^4 + 15*x^2 - 3
G(7,x) = 7*x^6 - 21*x^5 + 35*x^3 - 21*x

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, pp. 573-574.

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

A001489(n) = G(2n, 0) = -G(2n, 1). Cf. A081733.

Cf. A060096/A060097, A027641/A027642.

p := proc(n,x) local j,k; add(binomial(n,k)*add(binomial(k,j)*2^j*bernoulli(j), j=0..k-1)*x^(n-k),k=0..n) end;
seq(print(sort(p(n,x))),n=1..8); # Peter Luschny, Jul 07 2009

g[n_, x_] := Sum[ k Binomial[n, k] EulerE[k-1, 0] x^(n-k), {k, 1, n}]; Table[ CoefficientList[g[n, x], x] // Reverse, {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2013, after Peter Luschny *)

G(n)=subst(polcoeff(serlaplace(2*x*exp(x*y)/(exp(x)+1)),n),y,x)

E.g.f.: Sum_{n >= 1} G(n, x)*t^n/n! = 2*t*e^(x*t)/(1 + e^t).
G(n, x) = Sum_{k=1..n} k*C(n, k)* Euler(k-1, 0)*x^(n-k). - Peter Luschny, Jul 13 2009
G(n, x) = n*Euler(n-1,x) = Sum_{k=0..n} binomial(n,k)*Bernoulli(k)*2*(1-2^k)*x^(n-k), with the Euler polynomials Euler(n,x) (see A060096/A060097) and Bernoulli numbers A027641/A027642. See the Graham et al. reference, pp. 573-574, Exercise 7.52. - Wolfdieter Lang, Mar 13 2017

cp's OEIS Frontend

A319932 a(n) = (1/720)*n*(n - 10)*(n - 1)*(n^3 - 34*n^2 + 181*n - 144).

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A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + k*x/(1 + k*x^2/(1 + k*x^3/(1 + k*x^4/(1 + k*x^5/(1 + ...)))))).

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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

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A291701 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - x^5/(1 - ...))))).

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Examples

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A098434 Triangle read by rows: coefficients of Genocchi polynomials G(n,x); n times the Euler polynomials.

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Author

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Programs

Maple

Mathematica

PARI

Formula

A319932 a(n) = (1/720)n(n - 10)(n - 1)(n^3 - 34n^2 + 181n - 144).

A286932 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 + kx/(1 + kx^2/(1 + kx^3/(1 + kx^4/(1 + k*x^5/(1 + ...)))))).

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.