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 1-6 of 6 results.

A174480 Rectangular array of coefficients in successive iterations of x*exp(x), as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 23, 1, 1, 5, 28, 102, 104, 1, 1, 6, 45, 274, 861, 537, 1, 1, 7, 66, 575, 3400, 8598, 3100, 1, 1, 8, 91, 1041, 9425, 50734, 98547, 19693, 1, 1, 9, 120, 1708, 21216, 187455, 880312, 1270160, 136064, 1, 1, 10, 153, 2612, 41629
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2010

Keywords

Comments

Triangle A174485 forms a matrix that transforms a diagonal into an adjacent diagonal in this array.

Examples

			Form an array of coefficients in the iterations of x*exp(x), which begin:
n=1: [1, 1, 1/2!, 1/3!, 1/4!, 1/5!, 1/6!, ...];
n=2: [1, 2, 6/2!, 23/3!, 104/4!, 537/5!, 3100/6!, ...];
n=3: [1, 3, 15/2!, 102/3!, 861/4!, 8598/5!, 98547/6!, ...];
n=4: [1, 4, 28/2!, 274/3!, 3400/4!, 50734/5!, 880312/6!, ...];
n=5: [1, 5, 45/2!, 575/3!, 9425/4!, 187455/5!, 4367245/6!, ...];
n=6: [1, 6, 66/2!, 1041/3!, 21216/4!, 527631/5!, 15441636/6!, ...];
n=7: [1, 7, 91/2!, 1708/3!, 41629/4!, 1242892/5!, 43806175/6!, ...];
n=8: [1, 8, 120/2!, 2612/3!, 74096/4!, 2582028/5!, 106459312/6!, ...];
n=9: [1, 9, 153/2!, 3789/3!, 122625/4!, 4885389/5!, 230689017/6!, ...];
n=10:[1, 10, 190/2!, 5275/3!, 191800/4!, 8599285/5!, 457584940/6!,...];
...
This array begins with the above unreduced numerators for n >= 1, k >= 1.

Crossrefs

Cf. A174485, diagonals: A174481, A174482, A174483, A174484.
Cf. rows: A080108, A174493, A174494, A174495.

Programs

  • PARI
    {T(n, k)=local(F=x, xEx=x*exp(x+x*O(x^(k+1)))); for(i=1,n,F=subst(F, x, xEx));(k-1)!*polcoeff(F, k)}

Formula

T(n,k) = [x^k/(k-1)! ] G_{n}(x) where G_{n}(x) = G_{n-1}(x*exp(x)) with G_0(x)=x, for n>=1, k>=1.

A174481 a(n) = coefficient of x^n/(n-1)! in the (n-1)-th iteration of x*exp(x) for n>=1.

Original entry on oeis.org

1, 1, 6, 102, 3400, 187455, 15441636, 1776667928, 272145104736, 53540399628405, 13156413372354340, 3949011172491569316, 1421739781364268435576, 604701975767931070422939, 299969585267917154906689660
Offset: 1

Views

Author

Paul D. Hanna, Apr 09 2010

Keywords

Examples

			The initial n-th iterations of x*exp(x) begin:
n=0: (1)*x;
n=1: x + (1)*x^2 + x^3/2! + x^4/3! + x^5/4! + x^6/5! +...
n=2: x + 2*x^2 +(6)*x^3/2! + 23*x^4/3! + 104*x^5/4! + 537*x^6/5! +...
n=3: x + 3*x^2 +15*x^3/2! +(102)*x^4/3! +861*x^5/4! +8598*x^6/5! +...
n=4: x + 4*x^2 +28*x^3/2! +274*x^4/3! +(3400)*x^5/4! +50734*x^6/5! +...
n=5: x + 5*x^2 +45*x^3/2! +575*x^4/3! +9425*x^5/4! +(187455)*x^6/5!+...
n=6: x + 6*x^2 +66*x^3/2! +1041*x^4/3! +21216*x^5/4!+527631*x^6/5! + (15441636)*x^7/6! +...
This sequence starts with the above coefficients in parathesis.

Crossrefs

Cf. A174480, A174482, A174483, A174484.

Programs

  • PARI
    {a(n)=local(E=x*exp(x+x*O(x^n)), F=x); for(i=1, n-1, F=subst(F, x, E)); (n-1)!*polcoeff(F, n)}

A174483 a(n) = coefficient of x^n/(n-1)! in the (n+1)-th iteration of x*exp(x) for n>=1.

Original entry on oeis.org

1, 3, 28, 575, 21216, 1242892, 106459312, 12577403841, 1962856001440, 391431498879806, 97160350830990624, 29387077319612739025, 10642369538735639329912, 4547196797035053394680280
Offset: 1

Views

Author

Paul D. Hanna, Apr 09 2010

Keywords

Examples

			The initial n-th iterations of x*exp(x) begin:
n=1: x + x^2 + x^3/2! + x^4/3! + x^5/4! + x^6/5! +...
n=2: (1)*x +2*x^2 + 6*x^3/2! + 23*x^4/3! + 104*x^5/4! + 537*x^6/5! +...
n=3: x +(3)*x^2 +15*x^3/2! +102*x^4/3! +861*x^5/4! +8598*x^6/5! +...
n=4: x + 4*x^2 +(28)*x^3/2! +274*x^4/3! +3400*x^5/4! +50734*x^6/5! +...
n=5: x + 5*x^2 +45*x^3/2! +(575)*x^4/3! +9425*x^5/4! +187455*x^6/5! +...
n=6: x + 6*x^2 +66*x^3/2! +1041*x^4/3! +(21216)*x^5/4!+527631*x^6/5!+...
n=7: x + 7*x^2 +91*x^3/2! +1708*x^4/3! +41629*x^5/4! +(1242892)*x^6/5! +...
This sequence starts with the above coefficients in parenthesis.

Crossrefs

Cf. A174480, A174481, A174482, A174484.

Programs

  • PARI
    {a(n)=local(E=x*exp(x+x*O(x^n)), F=x); for(i=1, n+1, F=subst(F, x, E)); (n-1)!*polcoeff(F, n)}

A174484 a(n) = coefficient of x^n/(n-1)! in the (n+2)-th iteration of x*exp(x) for n>=1.

Original entry on oeis.org

1, 4, 45, 1041, 41629, 2582028, 230689017, 28145738365, 4504704373961, 916668654429870, 231318743221265869, 70928148561381638541, 25983184166531408190165, 11210928989636995091435576
Offset: 1

Views

Author

Paul D. Hanna, Apr 09 2010

Keywords

Examples

			The initial n-th iterations of x*exp(x) begin:
n=1: x + x^2 + x^3/2! + x^4/3! + x^5/4! + x^6/5! +...
n=2: x +2*x^2 + 6*x^3/2! + 23*x^4/3! + 104*x^5/4! + 537*x^6/5! +...
n=3: (1)*x +3*x^2 +15*x^3/2! +102*x^4/3! +861*x^5/4! +8598*x^6/5! +...
n=4: x +(4)*x^2 +28*x^3/2! +274*x^4/3! +3400*x^5/4! +50734*x^6/5! +...
n=5: x + 5*x^2 +(45)*x^3/2! +575*x^4/3! +9425*x^5/4! +187455*x^6/5! +...
n=6: x + 6*x^2 +66*x^3/2! +(1041)*x^4/3! +21216*x^5/4!+527631*x^6/5!+...
n=7: x + 7*x^2 +91*x^3/2! +1708*x^4/3! +(41629)*x^5/4! +1242892*x^6/5! +...
n=8: x + 8*x^2 +120*x^3/2! +2612*x^4/3! +74096*x^5/4!+(2582028)*x^6/5! +...
This sequence starts with the above coefficients in parenthesis.

Crossrefs

Cf. A174480, A174481, A174482, A174483.

Programs

  • PARI
    {a(n)=local(E=x*exp(x+x*O(x^n)), F=x); for(i=1, n+2, F=subst(F, x, E)); (n-1)!*polcoeff(F, n)}

A174485 Triangle of numerators T(n,k) in the matrix {T(n,k)/(n-k)!,n>=k>=0} that transforms diagonals of the array (A174480) of coefficients in successive iterations of x*exp(x).

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 70, 16, 3, 1, 1973, 308, 33, 4, 1, 94216, 11048, 810, 56, 5, 1, 6851197, 639972, 35325, 1672, 85, 6, 1, 706335064, 54671188, 2408568, 85904, 2990, 120, 7, 1, 98105431657, 6471586298, 236624733, 6741544, 176885, 4860, 161, 8, 1
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2010

Keywords

Examples

			Triangle T begins:
1;
1,1;
5,2,1;
70,16,3,1;
1973,308,33,4,1;
94216,11048,810,56,5,1;
6851197,639972,35325,1672,85,6,1;
706335064,54671188,2408568,85904,2990,120,7,1;
98105431657,6471586298,236624733,6741544,176885,4860,161,8,1;
17669939141440,1014487323984,31654735416,749040472,15706200,325368,7378,208,9,1;
...
Form a table of coefficients in iterations of x*exp(x), like so:
n=0: [1, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 1, 1/2!, 1/3!, 1/4!, 1/5!, 1/6!, ...];
n=2: [1, 2, 6/2!, 23/3!, 104/4!, 537/5!, 3100/6!, ...];
n=3: [1, 3, 15/2!, 102/3!, 861/4!, 8598/5!, 98547/6!, ...];
n=4: [1, 4, 28/2!, 274/3!, 3400/4!, 50734/5!, 880312/6!, ...];
n=5: [1, 5, 45/2!, 575/3!, 9425/4!, 187455/5!, 4367245/6!, ...];
n=6: [1, 6, 66/2!, 1041/3!, 21216/4!, 527631/5!+ 15441636/6!, ...];
n=7: [1, 7, 91/2!, 1708/3!, 41629/4!, 1242892/5!, 43806175/6!, ...];
n=8: [1, 8, 120/2!, 2612/3!, 74096/4!, 2582028/5!, 106459312/6!, ...];
...
and form matrix D from this triangle T by: D(n,k) = T(n,k)/(n-k)!,
then matrix D transforms diagonals in the above table as illustrated by:
D * A174481 = A174482, D * A174482 = A174483, D * A174483 = A174484,
where the diagonals begin:
A174481: [1, 1, 6/2!, 102/3!, 3400/4!, 187455/5!, ...];
A174482: [1, 2, 15/2!, 274/3!, 9425/4!, 527631/5!, ...];
A174483: [1, 3, 28/2!, 575/3!, 21216/4!, 1242892/5!, ...];
A174484: [1, 4, 45/2!, 1041/3!, 41629/4!, 2582028/5!, ...].

Crossrefs

Cf. columns: A174486, A174487, A174488, A174489.
Cf. A174480, A174481, A174482, A174483, A174484.

Programs

  • PARI
    {T(n, k)=local(F=x, xEx=x*exp(x+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, xEx)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (n-k)!*(P~*N~^-1)[n+1, k+1]}

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Examples

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Links

Crossrefs

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

Programs

  • Maple
    g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022
  • Mathematica
    Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]
  • PARI
    T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

Formula

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022
Showing 1-6 of 6 results.

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