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 1-9 of 9 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).

A113133 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 = 6.

Original entry on oeis.org

1, 1, 6, 72, 1332, 33264, 1040256, 38926656, 1692061488, 83688313536, 4638320578944, 284692939944192, 19169186341398912, 1404935464314299904, 111348880778746460160, 9489756817594314049536, 865470841829802331976448
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 6.
a(3) = 2*6^2 = 72.
a(4) = 6*3*72 + 1*6*6 = 1332.
a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.
a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.
G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5
+...
= x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5
+...).

Crossrefs

Cf. A008542, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113134(x=7), A113135(x=8).

Programs

  • Mathematica
    x=6;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-2,6*j+1))))))[n+1]
  • PARI
    a(n,x=6)=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} 6^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of sextuple factorial numbers (A008542).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of sextuple factorial numbers (A008542).

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 7, 3, 1, 0, 105, 36, 12, 4, 1, 0, 945, 249, 64, 18, 5, 1, 0, 10395, 2190, 441, 100, 25, 6, 1, 0, 135135, 23535, 3807, 691, 145, 33, 7, 1, 0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1
Offset: 0

Views

Author

Philippe Deléham, Oct 13 2005, Dec 20 2008

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, given by [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Examples

			Rows begin:
  1;
  0,       1;
  0,       1,      1;
  0,       3,      2,     1;
  0,      15,      7,     3,    1;
  0,     105,     36,    12,    4,    1;
  0,     945,    249,    64,   18,    5,   1;
  0,   10395,   2190,   441,  100,   25,   6,  1:
  0,  135135,  23535,  3807,  691,  145,  33,  7, 1;
  0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1;

Crossrefs

Columns: A000007, A001147, A034430; diagonals: A000012, A001477, A055998.
Cf. A084938, A111088, A112934, A168441.

Programs

  • Maple
    # Uses function PMatrix from A357368.
PMatrix(10, n -> doublefactorial(2*n-3)); # Peter Luschny, Oct 19 2022

Formula

T(n, k) = Sum_{j=0..n-k} T(n-1, k-1+j)*A111088(j).
Sum_{k=0..n} T(n, k) = A112934(n).
G.f.: 1/(1-xy/(1-x/(1-2x/(1-3x/(1-4x/(1-... (continued fraction). - Paul Barry, Jan 29 2009
Sum_{k=0..n} T(n,k)*2^(n-k) = A168441(n). - Philippe Deléham, Nov 28 2009

A232967 Irregular triangle read by rows: row n lists the rank sizes of the "electrical" poset EP_n of circular planar graphs with n boundary vertices.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 6, 14, 16, 10, 4, 1, 1, 10, 40, 85, 110, 97, 65, 35, 15, 5, 1, 1, 15, 90, 295, 609, 873, 948, 840, 636, 421, 246, 126, 56, 21, 6, 1, 1, 21, 175, 805, 2366, 4872, 7567, 9459, 10031, 9359, 7861, 6027, 4249, 2765, 1661, 917, 462, 210, 84, 28, 7, 1, 1, 28, 308, 1876, 7350, 20272, 42090, 69620, 96334, 115980, 125044, 123176, 112380, 95836, 76868, 58220, 41734, 28344, 18236, 11096, 6364, 3424, 1716, 792, 330, 120, 36, 8, 1
Offset: 1

Views

Author

N. J. A. Sloane, Dec 04 2013

Keywords

Comments

More than the usual number of terms are given, since the connection with A111088 seems rather mysterious.

Examples

			Triangle begins:
1,
1, 1,
1, 3, 3, 1,
1, 6, 14, 16, 10, 4, 1,
1, 10, 40, 85, 110, 97, 65, 35, 15, 5, 1,,
1, 15, 90, 295, 609, 873, 948, 840, 636, 421, 246, 126, 56, 21, 6, 1,
1, 21, 175, 805, 2366, 4872, 7567, 9459, 10031, 9359, 7861, 6027, 4249, 2765, 1661, 917, 462, 210, 84, 28, 7, 1,
1, 28, 308, 1876, 7350, 20272, 42090, 69620, 96334, 115980, 125044, 123176, 112380, 95836, 76868, 58220, 41734, 28344, 18236, 11096, 6364, 3424, 1716, 792, 330, 120, 36, 8, 1,
...

Links

Crossrefs

Row sums give A111088.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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