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 1-10 of 13 results. Next

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]}

A174482 a(n) = coefficient of x^n/(n-1)! in the n-th iteration of x*exp(x) for n>=1.

Original entry on oeis.org

1, 2, 15, 274, 9425, 527631, 43806175, 5060694920, 776717906529, 152926864265845, 37581193509020711, 11276280009364700628, 4057223684795928824281, 1724304353051995724792979
Offset: 1

Views

Author

Paul D. Hanna, Apr 09 2010

Keywords

Examples

			The initial n-th iterations of x*exp(x) begin:
n=1: (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: 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!+...
This sequence starts with the above coefficients in parathesis.

Crossrefs

Cf. A174480, A174481, A174483, A174484.

Programs

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

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)}

A174486 Column 0 of triangle A174485.

Original entry on oeis.org

1, 1, 5, 70, 1973, 94216, 6851197, 706335064, 98105431657, 17669939141440, 4006704580744601, 1117139031649249984, 375701872315954792093, 149988716080978525265776, 70129434038848683974552365
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2010

Keywords

Comments

Triangular matrix described by A174485 transforms diagonals of the array A174480 of coefficients of successive iterations of x*exp(x).

Crossrefs

Cf. A174480, A174485, A174487, A174488, A174489.

Programs

  • PARI
    {a(n,k=0)=local(F=x, xEx=x*exp(x+x*O(x^(n+k+1))), M, N, P, m=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!*(P~*N~^-1)[n+k+1, k+1]}

A174487 Column 1 of triangle A174485.

Original entry on oeis.org

1, 2, 16, 308, 11048, 639972, 54671188, 6471586298, 1014487323984, 203492881479464, 50842872702666524, 15484223252089602342, 5646860009850046968472, 2429577079632942917710580
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2010

Keywords

Comments

Triangular matrix described by A174485 transforms diagonals of the array A174480 of coefficients of successive iterations of x*exp(x).

Crossrefs

Cf. A174480, A174485, A174486, A174488, A174489.

Programs

  • PARI
    {a(n,k=1)=local(F=x, xEx=x*exp(x+x*O(x^(n+k+1))), M, N, P, m=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!*(P~*N~^-1)[n+k+1, k+1]}

A174488 Column 2 of triangle A174485.

Original entry on oeis.org

1, 3, 33, 810, 35325, 2408568, 236624733, 31654735416, 5532363865977, 1223887080470256, 334272773792556369, 110467177430468340408, 43442224822360939240629, 20048090531903711663566248
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2010

Keywords

Comments

Triangular matrix described by A174485 transforms diagonals of the array A174480 of coefficients of successive iterations of x*exp(x).

Crossrefs

Cf. A174480, A174485, A174486, A174487, A174489.

Programs

  • PARI
    {a(n,k=2)=local(F=x, xEx=x*exp(x+x*O(x^(n+k+1))), M, N, P, m=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!*(P~*N~^-1)[n+k+1, k+1]}

A174489 Column 3 of triangle A174485.

Original entry on oeis.org

1, 4, 56, 1672, 85904, 6741544, 749040472, 111786940612, 21558649749088, 5215883627856592, 1546429233541304456, 551278120123210461436, 232603216443181020788944, 114634034948809175011787176
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2010

Keywords

Comments

Triangular matrix described by A174485 transforms diagonals of the array A174480 of coefficients of successive iterations of x*exp(x).

Crossrefs

Cf. A174480, A174485, A174486, A174487, A174488.

Programs

  • PARI
    {a(n,k=3)=local(F=x, xEx=x*exp(x+x*O(x^(n+k+1))), M, N, P, m=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!*(P~*N~^-1)[n+k+1, k+1]}

A174493 a(n) = coefficient of x^n/(n-1)! in the 3-fold iteration of x*exp(x).

Original entry on oeis.org

1, 3, 15, 102, 861, 8598, 98547, 1270160, 18138601, 283754826, 4818884319, 88186786020, 1728395865021, 36091833338174, 799408841413051, 18708996086926272, 461095012437724881, 11931573394008790290
Offset: 1

Views

Author

Paul D. Hanna, Apr 17 2010

Keywords

Examples

			E.g.f.: x + 3*x^2 + 15*x^3/2! + 102*x^4/3! + 861*x^5/4! +...

Crossrefs

Cf. A174480, A080108, A174494, A174495, A174496.

Programs

  • PARI
    {a(n)=local(F=x, xEx=x*exp(x+x*O(x^n)));for(i=1,3,F=subst(F, x, xEx));(n-1)!*polcoeff(F, n)}
  • PARI
    {a(n)=sum(k=0,n-1,binomial(n-1,k)*sum(j=0,n-1-k,binomial(n-1-k,j)*(k+1)^j*(k+1+j)^(n-1-k-j)))}

Formula

a(n) = Sum_{k=0..n, j=0..n-k} C(n,k)*C(n-k,j)*(k+1)^j*(k+1+j)^(n-k-j).
O.g.f.: Sum_{n>=1} A080108(n)*x^n/(1-n*x)^n, where A080108(n) = [x^n/(n-1)! ] E(E(x)) and E(x) = x*exp(x).
Showing 1-10 of 13 results. Next

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