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.

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A385765 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x) - x^6*A'''''(x)).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 5172, 3739389, 9434483630, 63428037194102, 959222215928392076, 29009757539769286481866, 1608387988236777669667251772, 152866019594999736359695792369300, 23609086665918990295149462904374925800, 5671917808033245221993631555503554148332485
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A000108, A088716, A385762, A385763, A385764.
Cf. A385761.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/(1-x*A[x]-x^6*A'''''[x]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+sum(k=1, 5, stirling(5, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + 24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * a(k) * a(n-1-k).

A385831 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 32, 961, 64467, 8255248, 1808137854, 625644428013, 322212826476551, 235861774406899499, 236570361788785389414, 315585587694401993913716, 546279374467805677562555764, 1201815582876341559500261276952, 3301389061225358326490572037897646
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385832, A385833, A385834.
Cf. A385763.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j^3)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ).

A385832 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 56, 4705, 1218747, 765389596, 994245193386, 2390167881074445, 9797301213263859467, 64309492440202351088387, 643287882516349276270085850, 9420307945482704895570131173916, 195367768417628005309741727943311572, 5580484965405704420901774303244279908840
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385833, A385834.
Cf. A385764.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j^4)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ).

A112911 Triangle T, read by rows, such that the matrix inverse satisfies: [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0, with T(n,n)=1 for n>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 85, 44, 15, 4, 1, 621, 298, 96, 24, 5, 1, 5236, 2358, 729, 176, 35, 6, 1, 49680, 21154, 6327, 1492, 290, 48, 7, 1, 521721, 211100, 61380, 14220, 2725, 444, 63, 8, 1, 5994155, 2313030, 655944, 149812, 28425, 4590, 644, 80, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Comments

Matrix inverse square satisfies: [T^-2](3*n+2,n) = 0 for n>=0.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
14,8,3,1;
85,44,15,4,1;
621,298,96,24,5,1;
5236,2358,729,176,35,6,1;
49680,21154,6327,1492,290,48,7,1; ...
Matrix inverse T^-1 begins:
1;
-1,1;
-1,-2*1,1;
-3,-2*1,-3*1,1;
-14,-2*3,-3*1,-4*1,1;
-85,-2*14,-3*3,-4*1,-5*1,1;
-621,-2*85,-3*14,-4*3,-5*1,-6*1,1; ...
where [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.

Crossrefs

Cf. A088716 (column 0), A112912 (column 1), A112913 (column 2), A112914 (column 3).

Programs

  • PARI
    {T(n,k)=local(A=Mat(1),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1] );));A=B);return((A^-1)[n+1,k+1])}

A112912 Column 1 of triangle A112911.

Original entry on oeis.org

1, 2, 8, 44, 298, 2358, 21154, 211100, 2313030, 27566654, 354806116, 4903884712, 72444584732, 1139381007880, 19012236634968, 335560664081388, 6247230709277958, 122375974186267566, 2516528486416495240
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2005

Keywords

Crossrefs

Cf. A088716, A112911.

Programs

  • PARI
    {a(n)=local(A=Mat(1),B);for(m=2,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,B[i,j]=-j*(A^-1)[i-j,1]);));A=B);return((A^-1)[n+2,2])}

Formula

a(n) = Sum_{k=0..n-1} (n-k+1)*A088716(k)*a(n-k-1) for n>0 with a(0)=1.

A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -2, 9, -56, 425, -3726, 36652, -397440, 4695489, -59941550, 821711605, -12037503384, 187689245588, -3104186515976, 54295661153700, -1001685184237056, 19444296845046033, -396260414466644574, 8460628832978195683, -188898511962856879400
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2009

Keywords

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 + ...
(d/dx) (x/A(x)) = 1 - 2*x + 9*x^2 - 56*x^3 + 425*x^4 - 3726*x^5 + ...
1/A(x) = 1 - x + 3*x^2 - 14*x^3 + 85*x^4 + ... + (-1)^n*A088716(n)*x^n + ...
where a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,-2,9,-56,425,-3726,36652,-397440,4695489,...;
A^2: 1,2,(-3),14,-90,702,-6297,63144,-695886,8334822,...;
A^3: 1,3,(-3),(16),-108,870,-7997,81774,-915798,11116902,...;
A^4: 1,4,-2,(16),(-115),960,-9050,94368,-1073658,13204560,...;
A^5: 1,5,0,15,(-115),(996),-9630,102365,-1182690,14730890,...;
A^6: 1,6,3,14,-111,(996),(-9870),106890,-1253466,15804548,...;
A^7: 1,7,7,14,-105,973,(-9870),(108816),-1294412,16514162,...;
A^8: 1,8,12,16,-98,936,-9704,(108816),(-1312227),16931984,...;
A^9: 1,9,18,21,-90,891,-9426,107406,(-1312227),(17116900),...;
A^10:1,10,25,30,-80,842,-9075,104980,-1298625,(17116900),...; ...
where coefficients [x^n] A(x)^(n+1) and [x^n] A(x)^n are
enclosed in parenthesis and equal (n+1)*A158884(n) for n > 1:
[ -3,16,-115,996,-9870,108816,-1312227,17116900,...];
compare to A158884:
[1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,...]
and also to the logarithmic derivative of A158884:
[1,-3,16,-115,996,-9870,108816,-1312227,17116900,...].

Links

Crossrefs

Cf. A088716, A158884, variant: A158882.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(
      b(j)*b(n-j-1)*(j+1), j=0..n-1))
    end:
a:= n-> `if`(n=0, 1, -(-1)^n*n*b(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 18 2020
  • Mathematica
    m = 19; A[_] = 1;
Do[A[x_] = 1 + x*D[x/A[x], x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)
  • Maxima
    Composita(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Composita(n-i-1,k-1,F),i,0,n-k);
array(a, 10);
a[1]:1;
af(n):=a[n];
for n:2 thru 10 do a[n]:n*sum(Composita(n-1, k, af)*(-1)^k , k, 1, n-1);
makelist(af(n),n,1,10); /* Vladimir Kruchinin, Dec 01 2011 */
  • 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]);A[n+1]}

Formula

G.f. satisfies: A(x) = 1 + x*(d/dx)(x/A(x)) so that x^2*A'(x) = x*A(x) + A(x)^2 - A(x)^3.
a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
G.f.: A(x) = 1/(Sum_{n>=0} (-1)^n*A088716(n)*x^n), where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*(d/dx)(x*F(x)).
G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)*A158884(n) for n > 1.

A351798 a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (k+2) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 4, 31, 377, 6531, 152452, 4619130, 176631345, 8334329638, 476245005316, 32437793281489, 2597918907028430, 241796318654003869, 25886976434072903664, 3159556047500264255868, 436160347706069120482893, 67621917400663695356651589, 11700923494462411106797164208
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

Crossrefs

Cf. A000108, A088716.

Programs

  • Maple
    A351798 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add((1+k)*(2+k)*procname(k)*procname(n-k-1),k=0..n-1) ;
        %/2 ;
    end if;
end proc:
seq(A351798(n),n=0..30) ; # R. J. Mathar, Aug 19 2022
  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/2) Sum[(k + 1) (k + 2) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 2 x^2 A[x] A'[x] + x^3 A[x] A''[x]/2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

A385952 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+3,3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 59, 1309, 48790, 2840931, 244770680, 29887602613, 4993307581843, 1108754325139526, 319359741512132370, 116893982001130825135, 53422902443413341967604, 30024521959524315980717288, 20477109546794819263709728560, 16750490995674468051531269811269
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385953, A385954, A385955.
Cf. A385945.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, binomial(j+3, 3)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..3} binomial(3,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

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.

A376095 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^2 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 5, 54, 983, 26863, 1029188, 52747686, 3491367091, 290276997159, 29639219057133, 3648073361410412, 532858993269296500, 91147584892512564076, 18051321652239427195456, 4098339933686479506696526, 1057506667415381878759070811, 307764793378228160791205354175
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015083, A051893, A088716, A376096, A376097.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 3 x^2 A[x] A'[x] + x^3 A[x] A''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 3 * x^2 * A(x) * A'(x) + x^3 * A(x) * A''(x).
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