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 1-5 of 5 results.

A301930 G.f. A(x,y) satisfies: A(x,y) = x * (1 + y*A(x,y)*A'(x,y)) / (1 + A(x,y)*A'(x,y)), where A'(x,y) = d/dx A(x,y).

Original entry on oeis.org

1, -1, 1, 4, -7, 3, -25, 63, -52, 14, 200, -661, 808, -432, 85, -1890, 7754, -12586, 10090, -3989, 621, 20248, -99450, 201726, -216125, 128869, -40504, 5236, -240069, 1375831, -3354625, 4508559, -3604985, 1713731, -448122, 49680, 3102000, -20349633, 58049510, -94012374, 94504280, -60352776, 23900178, -5362906, 521721, -43226590, 319817454, -1046234664, 1985688420, -2408884136, 1936407600, -1031098592, 350561508, -69025155, 5994155
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2018

Keywords

Comments

Compare to: C(x) = x*(1 + 2*C(x)*C'(x)) / (1 + C(x)*C'(x)) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k = x + (-1 + y)*x^2 + (4 - 7*y + 3*y^2)*x^3 +  (-25 + 63*y - 52*y^2 + 14*y^3)*x^4 + (200 - 661*y + 808*y^2 - 432*y^3 + 85*y^4)*x^5 + (-1890 + 7754*y - 12586*y^2 + 10090*y^3 - 3989*y^4 + 621*y^5)*x^6 + ...
such that A = A(x,y) satisfies A = x*(1 + y*A*A')/(1 + A*A').
This triangle of coefficients T(n,k) in A(x,y) begins:
[1];
[-1, 1];
[4, -7, 3];
[-25, 63, -52, 14];
[200, -661, 808, -432, 85];
[-1890, 7754, -12586, 10090, -3989, 621];
[20248, -99450, 201726, -216125, 128869, -40504, 5236];
[-240069, 1375831, -3354625, 4508559, -3604985, 1713731, -448122, 49680];
[3102000, -20349633, 58049510, -94012374, 94504280, -60352776, 23900178, -5362906, 521721];
[-43226590, 319817454, -1046234664, 1985688420, -2408884136, 1936407600, -1031098592, 350561508, -69025155, 5994155]; ...
SPECIAL CASES.
G.f. C(x) of column 0 satisfies: C = x - C'*C^2, and begins C(x) = x - x^2 + 4*x^3 - 25*x^4 + 200*x^5 - 1890*x^6 +...
G.f. D(x) of the main diagonal satisfies: D = x + x*D'*D, and begins D(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + ...
At y = 2, the row polynomials evaluate to form the Catalan numbers:
1 = 1;
1 = -1 + 1*2;
2 = 4 + -7*2 + 3*2^2;
5 = -25 + 63*2 + -52*2^2 + 14*2^3;
14 = 200 + -661*2 + 808*2^2 + -432*2^3 + 85*2^4;
42 = -1890 + 7754*2 + -12586*2^2 + 10090*2^3 + -3989*2^4 + 621*2^5; ...
illustrating: C(2*n-1,n-1)/(2*n-1) = Sum_{k=0..n-1} T(n,k) * 2^k.
Note: when the g.f. A(x,y) is evaluated at y < 2 and y not= 1, the resulting power series in x will have negative coefficients somewhere in the expansion.

Links

Crossrefs

Cf. A088716, A182304, A301931, A301932, A301933.

Programs

  • PARI
    {T(n,k) = my(A=x); for(i=1,n, A = x*(1 + y*A*A')/(1 + A*A' +x*O(x^n))); polcoeff(polcoeff(A,n,x),k,y)}
/* Print as a triangle */
for(n=1,10,for(k=0,n-1, print1(T(n,k),", "));print(""))
/* Print as a flattened triangle: */
for(n=1,10, for(k=0,n-1, print1(T(n,k),", "); );)

Formula

Column 0 equals A088716 (signed).
Main diagonal equals A182304.
Row sums are zeros after the initial row.
Absolute row sums = A301931.
Sum_{k=0..n-1} T(n,k) * 2^k = C(2*n-1,n-1)/(2*n-1) = A000108(n-1) for n>=1.
Sum_{k=0..n-1} T(n,k) * 3^k = A301932(n) for n>=1.
Sum_{k=0..n-1} T(n,k) * 4^k = A301933(n) for n>=1.
Limit of largest real root of row polynomials converges to 2.

A245118 G.f. satisfies: A(x) = 1 + x*A(x)^3 / (A(x) - x*A'(x)).

Original entry on oeis.org

1, 1, 3, 15, 103, 879, 8799, 99711, 1250175, 17077215, 251414175, 3958086015, 66245028447, 1173340901695, 21914481618591, 430320455540223, 8861927054203743, 190989549802914495, 4299479144808402975, 100927457271249951615, 2466702562122306371295, 62678175995432599506495
Offset: 0

Views

Author

Paul D. Hanna, Jul 27 2014

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 879*x^5 + 8799*x^6 +...
The table of coefficients of x^k in A(x)^n begin:
n=1: [1,  1,  3,  15,  103,   879,   8799,   99711,  1250175, ...]
n=2: [1,  2,  7,  36,  245,  2054,  20199,  225384,  2789545, ...]
n=3: [1,  3, 12,  64,  435,  3597,  34806,  382590,  4674405, ...]
n=4: [1,  4, 18, 100,  683,  5592,  53340,  577960,  6971325, ...]
n=5: [1,  5, 25, 145, 1000,  8136,  76650,  819350,  9758835, ...]
n=6: [1,  6, 33, 200, 1398, 11340, 105730, 1116024, 13129335, ...]
n=7: [1,  7, 42, 266, 1890, 15330, 141736, 1478856, 17191251, ...]
n=8: [1,  8, 52, 344, 2490, 20248, 186004, 1920552, 22071459, ...]
n=9: [1,  9, 63, 435, 3213, 26253, 240069, 2455893, 27918000, ...]
n=10:[1, 10, 75, 540, 4075, 33522, 305685, 3102000, 34903110, ...]; ...
in which the diagonals illustrate the relation
[x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1
as follows:
[x^1] A(x)^2 = 2 = 2*[x^0] A(x)^3 = 2*1 ;
[x^2] A(x)^3 = 12 = 3*[x^1] A(x)^4 = 3*4 ;
[x^3] A(x)^4 = 100 = 4*[x^2] A(x)^5 = 4*25 ;
[x^4] A(x)^5 = 1000 = 5*[x^3] A(x)^6 = 5*200 ;
[x^5] A(x)^6 = 11340 = 6*[x^4] A(x)^7 = 6*1890 ;
[x^6] A(x)^7 = 141736 = 7*[x^5] A(x)^8 = 7*20248 ;
[x^7] A(x)^8 = 1920552 = 8*[x^6] A(x)^9 = 8*240069 ;
[x^8] A(x)^9 = 27918000 = 9*[x^7] A(x)^10 = 9*3102000 ; ...
RELATED SERIES.
Also, from the above table we can generate A182304:
[1/1, 2/2, 12/3, 100/4, 1000/5, 11340/6, 141736/7, 1920552/8, ...]
= [1, 1, 4, 25, 200, 1890, 20248, 240069, 3102000, 43226590, ...];
the g.f. of A182304 begins:
G(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +...
where G(x) = x + d/dx G(x)^3/3 and G(x) = A(G(x)).

Links

Crossrefs

Cf. A182304, A177384, A245768.

Programs

  • PARI
    /* From [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2): */
{a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=((#A)*Vec(Ser(A)^(#A+1))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From A(x) = 1 +  x*A(x)^3 / (A(x) - x*A'(x)): */
{a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^3/(A - x*A' +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1.
(2) A(x) = x/Series_Reversion(G(x)) where G(x) = x + d/dx G(x)^3/3 is the g.f. of A182304; thus, G(x) = A(G(x)).
a(n) ~ c * n! * n^4, where c = 0.005428317993266202636748034138... (same as for A177384). - Vaclav Kotesovec, Jul 28 2014

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

A385801 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3 - x^2*A(x)^2*A'(x)).

Original entry on oeis.org

1, 1, 5, 39, 385, 4438, 57375, 812720, 12428977, 203183595, 3525740946, 64607354042, 1245332551755, 25172891719156, 532300335665640, 11750901331656240, 270347716681495537, 6472208752084251545, 161016802667814325743, 4157582808082008519225
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A182304, A385806.
Cf. A385767.

Programs

  • Mathematica
    terms = 20; A[] = 0; Do[A[x] = 1/(1-x*A[x]^3-x^2*A[x]^2*A'[x]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)

Formula

a(0) = 1; a(n) = Sum_{i, j, k, l>=0 and i+j+k+l=n-1} (i+1) a(i) * a(j) * a(k) * a(l).

A385806 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^4 - x^2*A(x)^3*A'(x)).

Original entry on oeis.org

1, 1, 6, 56, 656, 8901, 134452, 2210098, 38972064, 730162940, 14436966166, 299765678868, 6512235121216, 147598065921110, 3482200915723080, 85360331346689846, 2170845829694670304, 57202138454461379820, 1559974375525184897080, 43985410131699875744400
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A182304, A385801.
Cf. A002294.

Programs

  • Mathematica
    terms = 20; A[] = 0; Do[A[x] = 1/(1-x*A[x]^4-x^2*A[x]^3*A'[x]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)

Formula

a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (i+1) a(i) * a(j) * a(k) * a(l) * a(m).
Showing 1-5 of 5 results.

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