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 1-4 of 4 results.

A158882 G.f. A(x) satisfies: [x^n] A(x)^n = [x^n] A(x)^(n-1) for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -13, 71, -461, 3447, -29093, 273343, -2829325, 31998903, -392743957, 5201061455, -73943424413, 1123596277863, -18176728317413, 311951144828863, -5661698774848621, 108355864447215063, -2181096921557783605
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2009

Keywords

Comments

After initial term, equals signed A003319 (indecomposable permutations).

Examples

			G.f.: A(x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 +-...
1/A(x) = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 +...+ (-1)^n*n!*x^n +...
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,(-1),3,-13,71,-461,3447,-29093,273343,-2829325,...;
A^2: 1,2,(-1),(4),-19,110,-745,5752,-49775,476994,-5016069,...;
A^3: 1,3, 0, (4),(-21),129,-910,7242,-64155,626319,-6685548,...;
A^4: 1,4, 2, 4, (-21),(136),-996,8152,-73811,733244,-7938186,...;
A^5: 1,5, 5, 5, -20, (136),(-1030),8650,-79925,807055,-8854741,...;
A^6: 1,6, 9, 8, -18, 132, (-1030),(8856),-83385,855010,-9500385,...;
A^7: 1,7,14,14, -14, 126, -1008, (8856),(-84861),882805,-9927890,...;
A^8: 1,8,20,24, -6, 120, -972, 8712, (-84861),(894928),-10180120,...;
A^9: 1,9,27,39,9,117,-927,8469,-83772,(894928),(-10291986),...;
A^10:1,10,35,60,35,122,-875,8160,-81890,885620,(-10291986),...; ...
where coefficients [x^n] A(x)^n and [x^n] A(x)^(n-1) are
enclosed in parenthesis and equal (-1)^n*n*A075834(n+1):
[ -1,4,-21,136,-1030,8856,-84861,894928,-10291986,128165720,...];
compare to A075834:
[1,1,1,2,7,34,206,1476,12123,111866,1143554,12816572,...]
and also to the logarithmic derivative of A075834:
[1,1,4,21,136,1030,8856,84861,894928,10291986,128165720,...].

Links

Crossrefs

Cf. A003319, A075834, A159311, variant: A158883.

Programs

  • Mathematica
    b[0] = 0; b[n_] := b[n] = n!-Sum[k!*b[n-k], {k, 1, n-1}]; a[0] = 1; a[n_] := (-1)^(n+1)*b[n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 07 2014, from 2nd formula *)
  • Maxima
    G(n,k):=(if n=k then 1 else if k=1 then (-sum(binomial(n-1,k-1)*G(n,k),k,2,n)) else sum(G(i+1,1)*G(n-i-1,k-1),i,0,n-k));
makelist(G(n,1),n,1,10); /* Vladimir Kruchinin, Mar 07 2014 */
  • PARI
    a(n)=polcoeff(1/sum(k=0,n,(-1)^k*k!*x^k +x*O(x^n)),n)
  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-2))-Vec(Ser(A)^(#A-1)))[ #A]);A[n+1]}

Formula

a(n) = (2-n) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
a(n) = (-1)^(n-1)*A003319(n) for n>=1.
G.f.: A(x) = 1/[Sum_{n>=0} (-1)^n*n!*x^n].
G.f. satisfies: [x^(n+1)] A(x)^n = (-1)^n*n*A075834(n+1) for n>=0.
From Sergei N. Gladkovskii, Jun 24 2012 to May 26 2013: (Start)
Continued fractions:
Let A(x) be the g.f., then A(x) = 1-x/U(0), where U(k) = x-1+x*k+(k+2)*x/U(k+1).
A(x) = 1/U(0), where U(k) = 1 - x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1)- 1/U(k+1))).
G.f.: U(0), where U(k)= 1 + x*(k+1)/(1 + x*(k+1)/U(k+1)).
G.f.: 2/(G(0) + 1), where G(k)= 1 - x*(k+1)/(1 - 1/(1 + 1/G(k+1))).
G.f.: x*G(0), where G(k)=1/x + 2*k + 1 - (k+1)^2/G(k+1).
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1))). (End)

A158884 G.f. A(x) satisfies: d/dx x*A(x) = 1+x + x*[d/dx log(A(x))].

Original entry on oeis.org

1, 1, -1, 4, -23, 166, -1410, 13602, -145803, 1711690, -21785618, 298370920, -4372151566, 68234087624, -1129894265272, 19788479904366, -365520041466291, 7103187300763530, -144897616964143050, 3096285550330959336
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - x^2 + 4*x^3 - 23*x^4 + 166*x^5 - 1410*x^6 +...
d/dx x*A(x) = 1 + 2*x - 3*x^2 + 16*x^3 - 115*x^4 + 996*x^5 - 9870*x^6 +...
d/dx log(A(x)) = 1 - 3*x + 16*x^2 - 115*x^3 + 996*x^4 - 9870*x^5 +...
Coefficients in powers A(x)^-n begin:
A(x)^-1: (1),-1,2,-7,36,-240,1926,-17815,184916,...;
A(x)^-2: (1),(-2),5,-18,90,-580,4525,-40946,417822,...;
A(x)^-3: 1,(-3),(9),-34,168,-1053,7997,-70776,709614,...;
A(x)^-4: 1,-4,(14),(-56),277,-1700,12594,-109032,1073658,...;
A(x)^-5: 1,-5,20,(-85),(425),-2571,18630,-157860,1526330,...;
A(x)^-6: 1,-6,27,-122,(621),(-3726),26492,-219912,2087658,...;
A(x)^-7: 1,-7,35,-168,875,(-5236),(36652),-298446,2782080,...;
A(x)^-8: 1,-8,44,-224,1198,-7184,(49680),(-397440),3639333,...; ...
where coefficients in parenthesis form A158883 and signed A088716
and A(x)^-1 (first row) is the g.f. of signed A088715.

Links

Crossrefs

Cf. A158883, A088715, A088716.

Programs

  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);Vec(Ser(A)^(n+1)/(n+1))[n+1]}

Formula

G.f. satisfies: x*A'(x) = A(x)*(1+x - A(x))/(A(x) - 1).
G.f.: A(x) = 1/G(-x) where G(x) is the g.f. of A088715.
G.f. satisfies: A(x/F(x)) = F(x) where F(x) is the g.f. of A158883.
G.f. satisfies: A(x*H(-x)) = H(-x) where H(x) is the g.f. of A088716.
G.f. satisfies: [x^n] 1/A(-x)^(n+2) = [x^(n+1)] 1/A(-x)^(n+2)/(n+2) = A088716(n+1).
a(n) ~ -(-1)^n * c * n! * n^2, where c = A238223 / exp(1) = 0.080179614624692622... - Vaclav Kotesovec, Nov 21 2017

A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + x*R(n,x)^n * [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 9, 1, 1, 0, 3, -56, 1, 1, 1, 0, -13, 425, 1, 1, 2, 0, 0, 71, -3726, 1, 1, 3, 3, -1, 0, -461, 36652, 1, 1, 4, 9, 0, 1, 0, 3447, -397440, 1, 1, 5, 18, 19, -12, 0, 0, -29093, 4695489, 1, 1, 6, 30, 72, 0, -14, 0, 0, 273343, -59941550
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2012

Keywords

Comments

The following property accounts for the zeros along the main diagonal.
The row g.f.s satisfy: [x^k] R(n,x)^(k-n+1) = [x^k] R(n,x)^(k-n) for k>=2
and thus when k=n we have [x^n] R(n,x) = [x^n] R(n,x)^0 = 0 for n>=2.

Examples

			Coefficients in the n-th row g.f., R(n,x), of this table begins:
n=0: [1, 1,-2,  9, -56,  425, -3726,  36652, -397440, 4695489, ...];
n=1: [1, 1,-1,  3, -13,   71,  -461,   3447,  -29093,  273343, ...];
n=2: [1, 1, 0,  0,   0,    0,     0,      0,       0,       0, ...];
n=3: [1, 1, 1,  0,  -1,    1,     0,      0,      -5,      27, ...];
n=4: [1, 1, 2,  3,   0,  -12,   -14,     43,      96,     -50, ...];
n=5: [1, 1, 3,  9,  19,    0,  -195,   -732,    -453,    6495, ...];
n=6: [1, 1, 4, 18,  72,  201,     0,  -4200,  -27984,  -91044, ...];
n=7: [1, 1, 5, 30, 175,  880,  3106,      0, -114485,-1124735, ...];
n=8: [1, 1, 6, 45, 344, 2451, 14946,  64522,       0,-3805692, ...];
n=9: [1, 1, 7, 63, 595, 5453, 45927, 331177, 1704795,       0, ...];
n=10:[1, 1, 8, 84, 944,10550,112336,1094604, 9157984,55095601, 0,...]; ...
in which the main diagonal is zeros for n>=2.
Initial row g.f.s are illustrated by the following.
R(0,x) = 1 + x*[d/dx x/R(0,x)] begins:
R(0,x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 +...
which satisfies: [x^k] R(0,x)^(k+1) = [x^k] R(0,x)^k for k>=2.
...
R(1,x) = 1 + x*R(1,x)*[d/dx x/R(1,x)] begins:
R(1,x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 + 3447*x^7 +...
which satisfies: [x^k] R(1,x)^k = [x^k] R(1,x)^(k-1) for k>=2.
...
R(2,x) = 1 + x*R(2,x)^2*[d/dx x/R(2,x)] is satisfied by:
R(2,x) = 1 + x,
which satisfies: [x^k] R(2,x)^(k-1) = [x^k] R(2,x)^(k-2) = 0 for k>=2.
...
R(3,x) = 1 + x*R(3,x)^3*[d/dx x/R(3,x)] begins:
R(3,x) = 1 + x + x^2 - x^4 + x^5 - 5*x^8 + 27*x^9 - 147*x^10 + 996*x^11 +...
which satisfies: [x^k] R(3,x)^(k-2) = [x^k] R(3,x)^(k-3) for k>=2.
...

Links

Crossrefs

Cf. A158883 (row 0), A158882 (row 1), A208897, A208898.

Programs

  • PARI
    {T(n,k)=local(ROWn=1+x+x*O(x^k));for(i=0,k,ROWn=1+x*ROWn^n*deriv(x/ROWn));polcoeff(ROWn,k)}
for(n=0,12,for(k=0,12,print1(T(n,k),","));print(""))

A177383 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^(n+2) for n>1 with A'(0)=A(0)=1.

Original entry on oeis.org

1, 1, -3, 18, -145, 1398, -15330, 186004, -2455893, 34903110, -529903726, 8547005820, -145847288730, 2624318836412, -49658727391620, 985921121129832, -20496980960236365, 445413655950148710, -10100793273684125430
Offset: 0

Views

Author

Paul D. Hanna, May 15 2010

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 18*x^3 - 145*x^4 + 1398*x^5 +...
Coefficients in the initial powers of A(x) begin:
[1,. 1, -3,. 18,. -145,. 1398,. -15330,. 186004,. -2455893,...];
[1,. 2, -5,. 30,. -245,. 2398,. -26670,. 327740,. -4376445,...];
[1,. 3,(-6), 37,. -309,. 3081,. -34806,. 433470,. -5855301,...];
[1,. 4,(-6),(40), -345,. 3516,. -40398,. 510120,. -6971325,...];
[1,. 5, -5, (40),(-360), 3761,. -43995,. 563460,. -7790675,...];
[1,. 6, -3,. 38, (-360),(3864), -46049,. 598266,. -8368635,...];
[1,. 7,. 0,. 35,. -350, (3864),(-46928), 618465,. -8751225,...];
[1,. 8,. 4,. 32,. -334,. 3792, (-46928),(627264), -8976609,...];
[1,. 9,. 9,. 30,. -315,. 3672,. -46284, (627264),(-9076320),...];
[1, 10, 15,. 30,. -295,. 3522,. -45180,. 620560, (-9076320),...];
where the above terms in parenthesis illustrate the property
that the coefficients of x^n in A(x)^(n+1) and A(x)^(n+2) are equal.
Also, the same terms derive A177384(n) = [x^n] A(x)^(n+1)/(n+1):
[1, 1, -2, 10, -72, 644, -6704, 78408, -1008480, 14065744, ...]
where g.f. G(x) of A177384 satisfies 1/x*d/dx[xG(x)]^2/2 = x+d/dx xG(x):
d/dx x*G(x) = 1 + 2*x - 6*x^2 + 40*x^3 - 360*x^4 + 3864*x^5 -...
d/dx [x*G(x)]^2/2 = x + 3*x^2 - 6*x^3 + 40*x^4 - 360*x^5 + 3864*x^6 -...

Links

Crossrefs

Cf. A177384, A158883, A182304.

Programs

  • PARI
    {a(n)=local(G=1+x+x*O(x^n),H);for(i=1,n,G=1+x/(G+x*deriv(G)+x*O(x^n))); H=x/serreverse(x*G);polcoeff(H,n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* From A(x) = 1 + x*(A(x) - x*A'(x)) / A(x)^3 : */
{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*(A-x*A')/(A^3 +x*O(x^n)));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + x * (A(x) - x*A'(x)) / A(x)^3. - Paul D. Hanna, Aug 04 2014
Let G(x) be the g.f. of A177384, then
. a(n) = [x^n] G(x)^(-n+1)/(-n+1) for n>1,
. A(x) = G(x/A(x)) so that A(x) = x/Series_Reversion(x*G(x))
where G(x) = 1 + x/(G(x) + x*G'(x)).
a(n) ~ c * (-1)^(n+1) * n! * n^4, where c = 0.0147556981601927885048672335828437... (same as for A182304). - Vaclav Kotesovec, Jul 28 2014
a(n) / A177384(n) ~ exp(1). - Vaclav Kotesovec, Mar 06 2020
Showing 1-4 of 4 results.

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