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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A218222 G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].

Original entry on oeis.org

1, 2, 12, 112, 1360, 19872, 335104, 6359040, 133560576, 3069007360, 76493880320, 2054400577536, 59136549994496, 1816392567062528, 59305340822814720, 2051451257317490688, 74958908119819812864, 2885480280276224311296, 116731741304854533111808
Offset: 1

Views

Author

Paul D. Hanna, Jan 31 2013

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 112*x^4 + 1360*x^5 + 19872*x^6 +...
Related series:
A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 272*x^5 + 3312*x^6 + 47872*x^7 + 794880*x^8 + 14840064*x^9 +...+ A112915(n-1)*x^n +...
d/dx A(x)^2 = 2*x + 12*x^2 + 112*x^3 + 1360*x^4 + 19872*x^5 +...

Links

Crossrefs

Cf. A000699, A112915, A185183.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      n*add(a(i)*a(n-i), i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 05 2020
  • Mathematica
    a[n_] := a[n] = If[n<2, 1, n*Sum[a[i]*a[n-i], {i, 1, n-1}]];
Array[a, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=x+x*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = 2^(n-1)*A088716(n-1) for n>=1, where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*d/dx[x*F(x)].
a(n) = n*A112915(n-1) for n>=1.
G.f.: x*d/dx x*G(x), where g.f. G(x) of A112915 satisfies: G(x) = 1 + x*(d/dx[x*G(x)])^2.
a(n) ~ c * n * 2^(n-1) * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017

A301305 G.f. L(x,y) satisfies: L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y)) where L'(x,y) = d/dx L(x,y), as a triangle read by rows.

Original entry on oeis.org

1, -1, 1, 3, -5, 2, -14, 33, -25, 6, 85, -261, 292, -140, 24, -621, 2363, -3516, 2546, -892, 120, 5236, -23872, 44537, -43405, 23228, -6444, 720, -49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040, 521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320, -5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880, 74701055, -578419961, 1998774636, -4056699966, 5351696394, -4791391134, 2945757656, -1226765624, 330797184, -52079040, 3628800
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2018

Keywords

Comments

Main diagonal equals the factorials.
Column 0 forms A088716, signed.
Rows sums are zeros after the initial row.
Absolute row sums equal A301388.

Examples

			G.f. L(x,y) = Sum_{n>=1, k=0..n-1} T(n,k) * x^n*y^k begins:
L(x,y) = x + (-1 + y)*x^2 + (3 - 5*y + 2*y^2)*x^3 + (-14 + 33*y - 25*y^2 + 6*y^3)*x^4 + (85 - 261*y + 292*y^2 - 140*y^3 + 24*y^4)*x^5 + (-621 + 2363*y - 3516*y^2 + 2546*y^3 - 892*y^4 + 120*y^5)*x^6 + (5236 - 23872*y + 44537*y^2 - 43405*y^3 + 23228*y^4 - 6444*y^5 + 720*y^6)*x^7 + (-49680 + 264860*y - 596396*y^2 + 733983*y^3 - 532095*y^4 + 226644*y^5 - 52356*y^6 + 5040*y^7)*x^8 + ...
where L = L(x,y) satisfies:
L = x*(1 + y*x*L') / (1 + x*L').
TRIANGLE.
This triangle of coefficients T(n,k) in L(x,y) begins:
1;
-1, 1;
3, -5, 2;
-14, 33, -25, 6;
85, -261, 292, -140, 24;
-621, 2363, -3516, 2546, -892, 120;
5236, -23872, 44537, -43405, 23228, -6444, 720;
-49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040;
521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320;
-5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880; ...
LIMITS.
In this triangle, the largest real root of the n-th row polynomial in y converges to the constant t = 2.845344903202547217277843362090557097661... (A301389).
RELATED SERIES.
exp(L(x,y)) = 1 + x + (-1 + 2*y)*x^2/2! + (13 - 24*y + 12*y^2)*x^3/3!  +  (-263 + 660*y - 540*y^2 + 144*y^3)*x^4/4!  +  (8381 - 26800*y + 31380*y^2 - 15840*y^3 + 2880*y^4)*x^5/5!  +  (-379409 + 1485870*y - 2280180*y^2 + 1706520*y^3 - 619200*y^4 + 86400*y^5)*x^6/6!  +  (22915369 - 106759128*y + 203726880*y^2 - 203269920*y^3 + 111449520*y^4 - 31691520*y^5 + 3628800*y^6)*x^7/7! + ...
satisfies: [x^n] exp(-n*L(x,y)) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp(-n*L(x,y)) for n>=1.

Links

Crossrefs

Cf. A088716, A301388, A301389.

Programs

  • PARI
    {T(n,k) = my(L=x); for(i=0,n, L = x*(1 + y*x*L')/(1 + x*L' +x*O(x^n)) ); polcoeff(polcoeff(L,n,x),k,y)}
for(n=1,12, for(k=0,n-1, print1(T(n,k),", "));print(""))

Formula

G.f. L(x,y) satisfies: [x^n] exp( -n * L(x,y) ) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp( -n * L(x,y) ) for n>=1.

A304402 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.

Original entry on oeis.org

1, 1, 9, 179, 5661, 249424, 14337039, 1035838044, 91867414241, 9833503227827, 1253246430314670, 187948018130914066, 32818034910964227439, 6608081830970361618546, 1520982783352578794866344, 397027611766464517915252056, 116698001659938095895315068553, 38375694701199964362412343063161
Offset: 0

Views

Author

Paul D. Hanna, May 25 2018

Keywords

Comments

Note: [x^n] exp( n * x*G(x) ) / G(x) = 0 for n>0 when G(x) is the g.f. of A088716.
It is remarkable that this sequence should consist entirely of integers.
What is the limit A304402(n) / A304400(n) ? Seems to be near 1.51...
A304402(n) / A304400(n) tends to 1.522998920075488836991600223419379... - Vaclav Kotesovec, Oct 06 2020

Examples

			O.g.f.: A(x) = 1 + x + 9*x^2 + 179*x^3 + 5661*x^4 + 249424*x^5 + 14337039*x^6 + 1035838044*x^7 + 91867414241*x^8 + 9833503227827*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) / A(x) begins:
n=0: [1, -1, -16, -972, -125952, -28275000, -9885939840, ...];
n=1: [1, 0, -15, -968, -125835, -28263864, -9883855835, ...];
n=2: [1, 3, 0, -860, -123456, -28073976, -9850185728, ...];
n=3: [1, 8, 65, 0, -104811, -26970576, -9680119083, ...];
n=4: [1, 15, 240, 3892, 0, -21937464, -9078485120, ...];
n=5: [1, 24, 609, 16528, 457173, 0, -7077136715, ...];
n=6: [1, 35, 1280, 49572, 2066880, 89033736, 0, ...];
n=7: [1, 48, 2385, 123880, 6839349, 411165624, 26124539077, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.
Terms along the secondary diagonal in the above table are divisible by the odd numbers: [1, 3/3, 65/5, 3892/7, 457173/9, 89033736/11, 26124539077/13, ...] = [1, 1, 13, 556, 50797, 8093976, 2009579929, ...].
RELATED SERIES.
exp( x*A(x) ) = 1 + x + 3*x^2/2! + 61*x^3/3! + 4537*x^4/4! + 702501*x^5/5! + 183891571*x^6/6! + 73567995313*x^7/7! + 42361186187601*x^8/8! + ...
The arithmetic inverse of the o.g.f. begins:
1/A(x) = 1 - x - 8*x^2 - 162*x^3 - 5248*x^4 - 235625*x^5 - 13730472*x^6 - 1001798042*x^7 - 89479215104*x^8 - 9627430506669*x^9 + ...

Links

Crossrefs

Cf. A304400, A088716.

Programs

  • PARI
    {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)^2*x*Ser(A) ) / Ser(A) )[m] );A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

a(n) ~ c * n!^2 * n^2, where c = 1.18365083976367345437640389636650727... - Vaclav Kotesovec, Oct 06 2020

A321086 O.g.f. A(x) satisfies: [x^n] exp(n*A(x)) * (1 - n*x - n*x^2) = 0, for n > 0.

Original entry on oeis.org

1, 2, 6, 32, 220, 1812, 17108, 180512, 2093760, 26396160, 358741328, 5223336288, 81079811280, 1336407320080, 23311138957200, 429063111959808, 8311760620707648, 169072470759431232, 3603666131945918144, 80327823251439861760, 1869212211081119135616, 45331401566332423284864, 1143967734536203174726784, 29996686272924492809481216, 816185909551276017516640000
Offset: 1

Views

Author

Paul D. Hanna, Oct 27 2018

Keywords

Comments

It is remarkable that this sequence should consist entirely of integers.
Compare to: [x^n] exp(n*G(x)) * (1 - n*x) = 0, for n > 0, when G(x) = x + x*G(x)*G'(x), where G(x)/x is the o.g.f. of A088716.

Examples

			O.g.f.: A(x) = x + 2*x^2 + 6*x^3 + 32*x^4 + 220*x^5 + 1812*x^6 + 17108*x^7 + 180512*x^8 + 2093760*x^9 + 26396160*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(-n*A(x)) * (1 - n*x - n*x^2) begins:
n=1: [1, 0, 1, 28, 729, 26416, 1321225, 87466716, ...];
n=2: [1, 0, 0, 32, 1200, 49152, 2569600, 172974720, ...];
n=3: [1, 0, -3, 0, 1089, 60408, 3509325, 246760776, ...];
n=4: [1, 0, -8, -80, 0, 49024, 3777280, 293683968, ...];
n=5: [1, 0, -15, -220, -2535, 0, 2848825, 291386100, ...];
n=6: [1, 0, -24, -432, -7056, -105984, 0, 208089216, ...];
n=7: [1, 0, -35, -728, -14175, -293048, -5733875, 0, ...];
n=8: [1, 0, -48, -1120, -24576, -590592, -15603200, -391709184, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
(a) Differential Equation.
O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x) where
A'(x) = 1 + 4*x + 18*x^2 + 128*x^3 + 1100*x^4 + 10872*x^5 + 119756*x^6 + ...
A(x)*A'(x) = x + 6*x^2 + 32*x^3 + 220*x^4 + 1812*x^5 + 17108*x^6 + 17108*x^7 + ...
so that A(x) - x*A(x)*A'(x) = x + x^2.
(b) Exponentiation.
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 985*x^4/4! + 32321*x^5/5! + 1544701*x^6/6! + 99637105*x^7/7! + 8257877489*x^8/8! + ...
exp(-A(x)) = 1 - x - 3*x^2/2! - 25*x^3/3! - 599*x^4/4! - 21681*x^5/5! - 1106939*x^6/6! - 74873737*x^7/7! - 6431021295*x^8/8! + ...

Crossrefs

Cf. A088716, A321085, A321087.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = -Vec( exp(m*x*Ser(A))*(1-m*x-m*x^2 +x^2*O(x^m)))[m+1]/m ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

O.g.f. A(x) satisfies: A(x) = x + x^2 + x*A(x)*A'(x).

A375393 a(0) = 1; a(n) = Sum_{k=0..n-1} (4*k+3) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 3, 30, 483, 10314, 268686, 8167068, 281975715, 10863651474, 461227101210, 21377716429860, 1073816307452430, 58106804389870500, 3370330005649001532, 208635817503306332088, 13731856676157543219747, 957698874584753026878306, 70562301536089812703526370, 5477354759932929856218644820
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2024

Keywords

Crossrefs

Cf. A000699, A088716, A151403, A215648, A376086, A376087.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(4 k + 3) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1 + 3 x A[x]^2 + 4 x^2 A'[x] A[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + 3 * x * A(x)^2 + 4 * x^2 * A'(x) * A(x).

A376086 a(0) = 1; a(n) = Sum_{k=0..n-1} (3*k+2) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 2, 14, 160, 2444, 45792, 1005480, 25169760, 705321200, 21841420384, 740194188032, 27243674154368, 1082259310732096, 46159435144505600, 2104195645965319680, 102113572703197079040, 5256795948307255075584, 286171738279517073904128, 16427146596936396844976640
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2024

Keywords

Crossrefs

Cf. A000699, A005159, A088716, A215648, A375393, A376087.

Programs

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

Formula

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

A376087 a(0) = 1; a(n) = Sum_{k=0..n-1} (4*k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 6, 65, 994, 19386, 456940, 12594465, 396969930, 14078044862, 554782989908, 24053551260186, 1138039204281236, 58353983394380500, 3223791843357228120, 190914111715994215905, 12065701995815379444954, 810602692757305194731094, 57688894099612173692496580
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 09 2024

Keywords

Crossrefs

Cf. A000699, A088716, A151403, A215648, A375393, A376086.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(4 k + 1) 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 + 4 x^2 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 + 4 * x^2 * A'(x) * A(x).

A376097 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^4 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 17, 1410, 364019, 228282823, 296324235500, 712075198644414, 2918094100584013255, 19151474626728425949663, 191553141880332262049655201, 2804913258838830873001491036584, 58168297154586087400230338311689652, 1661461159115675581245556180230933084340
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015085, A088716, A256020, A376095, A376096.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^4 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 15 x^2 A[x] A'[x] + 25 x^3 A[x] A''[x] + 10 x^4 A[x] A'''[x] + x^5 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 + 15 * x^2 * A(x) * A'(x) + 25 * x^3 * A(x) * A''(x) + 10 * x^4 * A(x) * A'''(x) + x^5 * A(x) * A''''(x).

A376134 a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^k * (k+1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, -1, -6, 17, 141, -660, -6688, 43837, 521755, -4412893, -60477282, 628119268, 9772644140, -120524236108, -2103803950976, 30068650440341, 582807287964375, -9477098158324107, -202143447363632090, 3686281848172281145, 85853256990102196221, -1735552985238117874788
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A088716, A090192, A105523, A376135, A376137.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-1)^k (k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 - x A[-x] + x^2 A'[-x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

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

A386229 G.f. A(x) satisfies A(x) = 1/( (1-x)^2 * (1 - x*A(x) - x^2*A'(x)) ).

Original entry on oeis.org

1, 3, 12, 70, 535, 4908, 51478, 600584, 7662285, 105684465, 1563183259, 24645719004, 412279514088, 7290426692472, 135862518564330, 2661378323466016, 54675576786754501, 1175673956931922257, 26411686616265112230, 618863341216409971750, 15101129008183181824938
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2025

Keywords

Crossrefs

Cf. A088716, A321087, A386230.
Cf. A386211.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1/((1-x)^2(1-x*A[x]-x^2*A'[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 16 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(i+1)*(1+sum(j=0, i-1, v[j+1]*v[i-j])/2)); v;

Formula

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