cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A159041 Triangle read by rows: row n (n>=0) gives the coefficients of the polynomial p(n,x) of degree n defined in comments.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, -25, -25, 1, 1, -56, 246, -56, 1, 1, -119, 1072, 1072, -119, 1, 1, -246, 4047, -11572, 4047, -246, 1, 1, -501, 14107, -74127, -74127, 14107, -501, 1, 1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1, 1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 03 2009

Keywords

Comments

Let E(n,k) (1 <= k <= n) denote the Eulerian numbers as defined in A008292. Then we define polynomials p(n,x) for n >= 0 as follows.
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*E(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*E(n+2,k+1)*x^k ).
For example,
p(0,x) = (1-x)/(1-x) = 1,
p(1,x) = (1-x^2)/(1-x) = 1 + x,
p(2,x) = (1 - 11*x + 11*x^2 - x^3)/(1-x) = 1 - 10*x + x^2,
p(3,x) = (1 - 26*x + 26*x^3 - x^4)/(1-x) = 1 - 25*x - 25*x^2 + x^3,
p(4,x) = (1 - 57*x + 302*x^2 - 302*x^3 + 57*x^3 + x^5)/(1-x)
= 1 - 56*x + 246*x^2 - 56*x^3 + x^4.
More generally, there is a triangle-to-triangle transformation U -> T defined as follows.
Let U(n,k) (1 <= k <= n) be a triangle of nonnegative numbers in which the rows are symmetric about the middle. Define polynomials p(n,x) for n >= 0 by
p(n,x) = (1/(1-x)) * ( Sum_{k=0..floor(n/2)} (-1)^k*U(n+2,k+1)*x^k + Sum_{k=ceiling((n+2)/2)..n+1} (-1)^(n+k)*U(n+2,k+1)*x^k ).
The n-th row of the new triangle T(n,k) (0 <= k <= n) gives the coefficients in the expansion of p(n+2).
The new triangle may be defined recursively by: T(n,0)=1; T(n,k) = T(n,k-1) + (-1)^k*U(n+2,k) for 1 <= k <= floor(n/2); T(n,k) = T(n,n-k).
Note that the central terms in the odd-numbered rows of U(n,k) do not get used.
The following table lists various sequences constructed using this transform:
Parameter Triangle Triangle Odd-numbered
m U T rows
0 A007312 A007312 A034870
1 A008292 A159041 A171692
2 A060187 A225356 A225076
3 A142458 A225433 A225398
4 A142459 A225434 A225415

Examples

			Triangle begins as follows:
  1;
  1,     1;
  1,   -10,      1;
  1,   -25,    -25,        1;
  1,   -56,    246,      -56,       1;
  1,  -119,   1072,     1072,    -119,       1;
  1,  -246,   4047,   -11572,    4047,    -246,        1;
  1,  -501,  14107,   -74127,  -74127,   14107,     -501,      1;
  1, -1012,  46828,  -408364,  901990, -408364,    46828,  -1012,     1;
  1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035, 1;

Links

Crossrefs

Cf. A007312, A008292, A034870, A060187, A142458, A142459, A159041, A171692, A225076, A225356, A225398, A225415, A225433, A225434.

Programs

  • Maple
    A008292 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 or k = n then 1; else k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1) ; end if; end proc:
# row n of new triangle T(n,k) in terms of old triangle U(n,k):
p:=proc(n) local k; global U;
simplify( (1/(1-x)) * ( add((-1)^k*U(n+2,k+1)*x^k,k=0..floor(n/2)) + add((-1)^(n+k)*U(n+2,k+1)*x^k, k=ceil((n+2)/2)..n+1 )) );
end;
U:=A008292;
for n from 0 to 6 do lprint(simplify(p(n))); od: # N. J. A. Sloane, May 11 2013
A159041 := proc(n, k)
    if k = 0 then
        1;
    elif k <= floor(n/2) then
        A159041(n, k-1)+(-1)^k*A008292(n+2, k+1) ;
    else
        A159041(n, n-k) ;
    end if;
end proc: # R. J. Mathar, May 08 2013
  • Mathematica
    A[n_, 1] := 1;
A[n_, n_] := 1;
A[n_, k_] := (n - k + 1)A[n - 1, k - 1] + k A[n - 1, k];
p[x_, n_] = Sum[x^i*If[i == Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], (-1)^i*A[n, i], -(-1)^(n - i)*A[n, i]]], {i, 0, n}]/(1 - x);
Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}];
Flatten[%]
  • Sage
    def A008292(n,k): return sum( (-1)^j*(k-j)^n*binomial(n+1,j) for j in (0..k) )
@CachedFunction
def T(n,k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return T(n,k-1) + (-1)^k*A008292(n+2,k+1)
    else: return T(n,n-k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = T(n, k-1) + (-1)^k*A008292(n+2, k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1. - R. J. Mathar, May 08 2013

Extensions

Edited by N. J. A. Sloane, May 07 2013, May 11 2013

A050393 Reversion of partitions into distinct parts A000009.

Original entry on oeis.org

1, -1, 0, 3, -7, 3, 31, -105, 101, 419, -1971, 2923, 5800, -40388, 81147, 64075, -854408, 2204543, -56096, -18070916, 58866158, -38939227, -371701743, 1544696638, -1870286829, -7166094999, 39743193694, -68677654555
Offset: 1

Views

Author

Christian G. Bower, Nov 15 1999

Keywords

Links

Crossrefs

Cf. A000009, A007312, A050394.

Programs

  • Mathematica
    InverseSeries[QPochhammer[-1, x]/2 + O[x]^20][[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)

Formula

G.f. A(x) satisfies: A(x) = -1 + (1 + x) * Product_{k>=2} 1/(1 + A(x)^k). - Ilya Gutkovskiy, Apr 23 2020

A066398 Reversion of g.f. (with constant term included) for partition numbers.

Original entry on oeis.org

1, -1, 0, 2, -3, 0, 5, 0, -21, 14, 117, -342, 210, 935, -2565, 1864, 2751, -3945, -8074, 4046, 108927, -333832, 246895, 887040, -2764795, 3062749, -1372098, 4775900, -9367698, -55130625, 299939766, -537241936, -140898285, 2464380030, -4060507784, 193070394
Offset: 0

Views

Author

N. J. A. Sloane, Dec 25 2001

Keywords

Comments

See A301624 for the corresponding series reversion for the plane partition numbers A000219. - Peter Bala, Feb 09 2020

Links

Crossrefs

Cf. A000041, A000203, A007312, A301624.

Programs

  • Maple
    with(numtheory):
Order := 36:
Gser := solve(series(x*exp(add(sigma[1](n)*x^n/n, n = 1..35)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..35); # Peter Bala, Feb 09 2020
  • Mathematica
    nmax = 34; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Product[ 1 - x^k*A[x]^k, {k, 1, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

Formula

The o.g.f. A(x) = 1 - x + 2*x^3 - 3*x^4 + 5*x^6 - ... satisfies [x^n](1/A(x))^n = sigma(n) = A000203(n) for n >= 1. - Peter Bala, Aug 23 2015
G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k). - Ilya Gutkovskiy, Mar 21 2018

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2009

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...

Crossrefs

Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.

Programs

  • Mathematica
    InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*Derivative[0, 0, 1][ QPolyGamma][0, 1, s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)
  • PARI
    {a(n)=polcoeff(serreverse(x*eta(x+x*O(x^n))^3),n)}

Formula

G.f. A(x) satisfies:
(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution cube of A171804 (with offset).
a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017

Extensions

More terms from Vladimir Reshetnikov, Nov 21 2016

A291489 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^k)^k.

Original entry on oeis.org

1, -2, 3, 2, -41, 196, -541, 229, 7235, -48228, 175956, -254933, -1575661, 14909191, -67194669, 153944915, 292516673, -4968647665, 27275432639, -82747735226, 3883854725, 1660136515050, -11302429310683, 42362000190568, -53376259124482, -520085199830413, 4671353423344131
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A026007.

Links

Crossrefs

Cf. A007312, A026007, A050393, A176025.

Programs

  • Mathematica
    nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Product[(1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 27; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x], x]]

Formula

G.f. A(x) satisfies: -1 + Product_{k>=1} (1 + A(x)^k)^k = x.

A334315 E.g.f. A(x) satisfies: A(x) = x - Sum_{k>=2} p(k) * A(x)^k / k!, where p = A000041 (partition numbers).

Original entry on oeis.org

1, -2, 9, -65, 653, -8432, 133188, -2488450, 53683569, -1313214351, 35916970957, -1086055854233, 35975402985863, -1295514629022924, 50391598721116365, -2105485003413499952, 94047072252968125326, -4472183077495496587696, 225565085807090517308839
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 22 2020

Keywords

Comments

Exponential reversion of A000041 (partition numbers).

Links

Crossrefs

Cf. A000041, A007312, A050394, A066398.

Programs

  • Mathematica
    nmax = 19; CoefficientList[InverseSeries[Series[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest

A291488 Expansion of the series reversion of -1 + Product_{k>=1} 1/(1 - x^k)^k.

Original entry on oeis.org

1, -3, 12, -58, 318, -1896, 11966, -78595, 531486, -3674324, 25845131, -184348434, 1330147092, -9690872427, 71189146313, -526703176813, 3921274277132, -29354616797397, 220824254874928, -1668453804382315, 12655766723174710, -96340024533522759, 735747052686408916, -5635489764030599334
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A000219.

Links

Crossrefs

Cf. A000219, A007312, A050393, A176025.

Programs

  • Mathematica
    nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[-1 + Exp[Sum[DivisorSigma[2, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x], x]]

Formula

G.f. A(x) satisfies: -1 + Product_{k>=1} 1/(1 - A(x)^k)^k = x.

A291645 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^(k^2)).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, 4, 9, 4, -23, -78, -78, 132, 694, 1088, -443, -6169, -13452, -4646, 52247, 155891, 143796, -391672, -1715015, -2481013, 2107735, 17836000, 35704800, 3037215, -172386166, -465009936, -338007604, 1487272659, 5624864403, 7125599375, -10208041482
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A033461.

Links

Crossrefs

Cf. A007312, A033461, A050393, A291489.

Programs

  • Mathematica
    nmax = 36; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1 + x^k^2, {k, 1, nmax}], {x, 0, nmax}], x], x]]

Formula

G.f. A(x) satisfies: -1 + Product_{k>=1} (1 + A(x)^(k^2)) = x.

A291646 Expansion of the series reversion of -1 + Product_{k>=1} (1 + x^(2*k-1)).

Original entry on oeis.org

1, 0, -1, -1, 2, 6, -1, -29, -32, 108, 311, -185, -1991, -1590, 9468, 22163, -26645, -170511, -70359, 955734, 1755790, -3561052, -16020532, 309754, 102695477, 141637053, -463468990, -1567907433, 806541136, 11367276801, 10768399120, -59447130815, -155142592628, 172852194214, 1273466836673
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Reversion of g.f. (with constant term omitted) for A000700.

Links

Crossrefs

Cf. A000700, A007312, A050393, A291489.

Programs

  • Mathematica
    nmax = 35; Rest[CoefficientList[InverseSeries[Series[-1 + Product[1 + x^(2 k - 1), {k, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 35; Rest[CoefficientList[InverseSeries[Series[-1 + QPochhammer[x^2]^2/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x], x]]

Formula

G.f. A(x) satisfies: -1 + Product_{k>=1} (1 + A(x)^(2*k-1)) = x.

A291695 Expansion of the series reversion of Sum_{i>=1} x^i/(1 - x^i) / Product_{j>=1} (1 - x^j).

Original entry on oeis.org

1, -3, 12, -57, 304, -1757, 10746, -68450, 449274, -3016645, 20618317, -142946735, 1002722249, -7103064540, 50738237140, -365049115546, 2642981328372, -19241453032254, 140770867457795, -1034409857616986, 7631075823632553, -56497364856268721, 419641611512419630, -3126180409889288924
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

Comments

Reversion of g.f. for A006128.

Links

Crossrefs

Cf. A006128, A007312, A050393, A176025.

Programs

  • Mathematica
    nmax = 24; Rest[CoefficientList[InverseSeries[Series[Sum[x^i/(1 - x^i), {i, 1, nmax}] / Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x], x]]
nmax = 24; Rest[CoefficientList[InverseSeries[Series[(Log[1-x] + QPolyGamma[0, 1, x]) / (Log[x]*QPochhammer[x]), {x, 0, nmax}], x], x]] (* Vaclav Kotesovec, Apr 21 2020 *)

Formula

G.f. A(x) satisfies: Sum_{i>=1} A(x)^i/(1 - A(x)^i) / Product_{j>=1} (1 - A(x)^j) = x.
G.f. A(x) satisfies: Sum_{i>=1} i*A(x)^i / Product_{j=1..i} (1 - A(x)^j) = x.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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