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.

A113129 Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 6, 0, 0, 0, 10, 24, 0, 0, 0, 4, 82, 120, 0, 0, 0, 0, 84, 672, 720, 0, 0, 0, 0, 27, 1236, 5820, 5040, 0, 0, 0, 0, 0, 930, 16328, 54288, 40320, 0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880, 0, 0, 0, 0, 0, 0, 12452, 396528, 2775432
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Comments

Let R(m,n,k), 0<=k<=n, the Riordan array (1,x*g(x)) where g(x) is g.f. of the m-fold factorials . Then R(m,n,k) = R(m,n-1,k-1) + Sum_{j, 0<=j<=n-1-k} R(m,n-1,k+j)*P_m(j), R(m,n,0) = 0^n and R(m,0,k) = 0 if k>n.

Examples

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 6;
.0, 0, 0, 10, 24;
.0, 0, 0, 4, 82, 120;
.0, 0, 0, 0, 84, 672, 720;
.0, 0, 0, 0, 27, 1236, 5820, 5040;
.0, 0, 0, 0, 0, 930, 16328, 54288, 40320;
.0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880;
.0, 0, 0, 0, 0, 0, 12452, 396528, 2775432, 6003360, 362880;
.0, 0, 0, 0, 0, 0, 2830, 38732, 7057308, 37831752, 71019360, 39916800;

Crossrefs

Cf. A000007, A000108, A000142, A000699.
R(m, n, k) : A097805 (m=0), A084938 (m=1), A111106 (m=2), A113333 (column sums).

Formula

P_0(x) = 1, P_1(x) = x, P_2(x) = 2*x^2, P_ n(x) = n*x*P_(n-1)(x) + Sum_{j, 1<=j<=n-1} j*P_j(x)*P_(n-1-j)(x).
P_n(x) = Sum_{k, 0<=k<=n} T(n, k)*x^k.
P_n(0) = A000007(n).
P_n(x) = A075834(n+1), A111088(n+1), A113130(n+1), A113131(n+1), A113132(n+1), A113133(n+1), A113134(n+1), A113135(n+1) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
P_n(-1) = (-1)^n*A000108(n), signed Catalan numbers.
T(n, n) = n! = A000142(n).
T(2*n+1, n+1) = A000699(n+1) (number of irreducible diagrams with 2n+2 nodes).
T(2*n+2, n+2) = A113332(n) = A000699(n+2)*(2*n+3)*(n+2)/(3*(n+1)).

Extensions

Corrected by Philippe Deléham, Dec 18 2008

A113134 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 7.

Original entry on oeis.org

1, 1, 7, 98, 2107, 61054, 2215094, 96203268, 4856212179, 279081882086, 17981777803682, 1283631249683804, 100557420457355358, 8577121056958121836, 791318123914138366924, 78521346319092948749576
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 7.
a(3) = 2*7^2 = 98.
a(4) = 7*3*98 + 1*7*7 = 2107.
a(5) = 7*4*2107 + 1*7*98 + 2*98*7 = 61054.
a(6) = 7*5*61054 + 1*7*2107 + 2*98*98 + 3*2107*7 = 2215094.
G.f.: A(x) = 1 + x + 7*x^2 + 98*x^3 + 2107*x^4 + 61054*x^5
+...
= x/series_reversion(x + x^2 + 8*x^3 + 120*x^4 + 2640*x^5
+...).

Crossrefs

Cf. A045754, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113135(x=8).

Programs

  • Mathematica
    x=7;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 16}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,7*j+1))))))[n+1]
  • PARI
    a(n,x=7)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 7^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 7-fold factorials.
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 7-fold factorials.

A113135 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 8.

Original entry on oeis.org

1, 1, 8, 128, 3136, 103424, 4270080, 211107840, 12135936000, 794618298368, 58355305676800, 4749550536359936, 424336070117163008, 41287521140173963264, 4346005245162898325504, 492102089936714946576384
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 8.
a(3) = 2*8^2 = 128.
a(4) = 8*3*128 + 1*8*8 = 3136.
a(5) = 8*4*3136 + 1*8*128 + 2*128*8 = 103424.
a(6) = 8*5*103424 + 1*8*3136 + 2*128*128 + 3*3136*8 = 4270080
G.f.: A(x) = 1 + x + 8*x^2 + 128*x^3 + 3136*x^4 + 103424*x^5 +...
= x/series_reversion(x + x^2 + 9*x^3 + 153*x^4 + 3825*x^5 +...).

Crossrefs

Cf. A045755, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7).

Programs

  • Mathematica
    x=8;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 16}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,8*j+1))))))[n+1]
  • PARI
    a(n,x=8)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 8^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of 8-fold factorials.
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of 8-fold factorials.

A113130 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 3.

Original entry on oeis.org

1, 1, 3, 18, 171, 2214, 35910, 694980, 15567795, 395396478, 11218141170, 351527039676, 12056563337598, 449255267318844, 18074052522890604, 780881956274215944, 36062953309417344579, 1772992806860541951342
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 3.
a(3) = 2*3^2 = 18.
a(4) = 3*3*18 + 1*3*3 = 171.
a(5) = 3*4*171 + 1*3*18 + 2*18*3 = 2214.
a(6) = 3*5*2214 + 1*3*171 + 2*18*18 + 3*171*3 = 35910.
G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 171*x^4 + 2214*x^5 +...
= x/series_reversion(x + x^2 + 4*x^3 + 28*x^4 + 280*x^5 +...).

Crossrefs

Cf. A007559, A075834(x=1), A111088(x=2), A113131(x=4), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

Programs

  • Mathematica
    x=3;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 18}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,3*j+1))))))[n+1]
  • PARI
    {a(n,x=3)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,
x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))}

Formula

a(n+1) = Sum{k, 0<=k<=n} 3^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of triple factorials (A007559).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007559).

A113131 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 4.

Original entry on oeis.org

1, 1, 4, 32, 400, 6784, 144128, 3658752, 107686656, 3599697920, 134617038848, 5567255822336, 252278661832704, 12431395516383232, 661885541595873280, 37869659304097218560, 2317293119684500193280, 151022143036329696952320
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 4.
a(3) = 2*4^2 = 32.
a(4) = 4*3*32 + 1*4*4 = 400.
a(5) = 4*4*400 + 1*4*32 + 2*32*4 = 6784.
a(6) = 4*5*6784 + 1*4*400 + 2*32*32 + 3*400*4 = 144128.
G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 400*x^4 + 6784*x^5 +...
= x/series_reversion(x + x^2 + 5*x^3 + 45*x^4 + 585*x^5 +...).

Crossrefs

Cf. A007696, A075834(x=1), A111088(x=2), A113130(x=3), A113132(x=5), A113133(x=6), A113134(x=7), A113135(x=8).

Programs

  • Mathematica
    x=4;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 18}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,4*j+1))))))[n+1]
  • PARI
    a(n,x=4)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 4^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quartic factorials (A007696).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of triple factorials (A007696).

A113132 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 5.

Original entry on oeis.org

1, 1, 5, 50, 775, 16250, 426750, 13402500, 488566875, 20249281250, 939823431250, 48278138937500, 2719288331093750, 166652371531562500, 11040797013538437500, 786338134640203125000, 59916445436152444921875
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 5.
a(3) = 2*5^2 = 50.
a(4) = 5*3*50 + 1*5*5 = 775.
a(5) = 5*4*775 + 1*5*50 + 2*50*5 = 16250.
a(6) = 5*5*16250 + 1*5*775 + 2*50*50 + 3*775*5 = 426750.
G.f.: A(x) = 1 + x + 5*x^2 + 50*x^3 + 775*x^4 + 16250*x^5 +...
= x/series_reversion(x + x^2 + 6*x^3 + 66*x^4 + 1056*x^5
+...).

Crossrefs

Cf. A008548, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113133(x=6), A113134(x=7), A113135(x=8).

Programs

  • Mathematica
    x=5;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 17}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,5*j+1))))))[n+1]
  • PARI
    a(n,x=5)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 5^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of quintic factorials (A008548).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of quintic factorials (A008548).
Showing 1-6 of 6 results.

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