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.

A113144 Row 3 of table A113143; equal to INVERT of triple (or 3-fold) factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 7, 41, 364, 4409, 67573, 1248626, 26948347, 664414997, 18409263772, 566018365445, 19117946453041, 703533848468330, 28013710891743007, 1199943043040160401, 55013996422974758476, 2687888298887895948065, 139414898768304344206141
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 7*x^3 + 41*x^4 + 364*x^5 + 4409*x^6
+...
= 1/(1 - x - x^2 - 4*x^3 - 28*x^4 -...- A007559(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A007559 (3-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,3*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 3^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A007559(n-k).
G.f.: 1/(Q(0)-x) where Q(k) = 1 - x*(3*k+1)/( 1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 21 2013

A113145 Row 4 of table A113143; equal to INVERT of quartic (or 4-fold) factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 8, 60, 708, 11428, 232756, 5704964, 163192820, 5331728964, 195776203764, 7978838333188, 357313060904692, 17438518614448580, 921145685670017012, 52355425184381107332, 3185815887918686343924, 206633438251087758833476
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 708*x^5 +...
= 1/(1 - x - x^2 - 5*x^3 - 45*x^4 -...- A007696(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A007696 (4-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,4*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 4^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A007696(n-k).
G.f.: 1/(T(0) - x) where T(k) = 1 - x*(4*k+1)/(1 - x*(4*k+4)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013

A113146 Row 5 of table A113143; equal to INVERT of quintic (or 5-fold) factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, 1270110048234, 64470498348983, 3596569233141701, 218698213338646702, 14395754017090902609, 1019782749198898131883, 77351848007810972904826
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 9*x^3 + 83*x^4 + 1226*x^5 +...
= 1/(1 - x - x^2 - 6*x^3 - 66*x^4 -...- A008548(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A008548 (5-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,5*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 5^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A008548(n-k).

A113147 Row 6 of table A113143; equal to INVERT of 6-fold factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, 5966767051354, 362558298692270, 24214789406313442, 1763062297639690790, 138975554045857840570, 11790733617760291994990, 1071215297856049456744642
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 10*x^3 + 110*x^4 + 1954*x^5 +...
= 1/(1 - x - x^2 - 7*x^3 - 91*x^4 -...- A008542(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A008542 (6-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,6*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 6^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A008542(n-k).

A113148 Row 7 of table A113143; equal to INVERT of 7-fold factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 11, 141, 2928, 82597, 2925973, 124502114, 6179425823, 350316271761, 22326710345256, 1579953165170881, 122905129550802985, 10423661531476766834, 957176457621821573987, 94608465923392572536421
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 11*x^3 + 141*x^4 + 2928*x^5 +...
= 1/(1 - x - x^2 - 8*x^3 - 120*x^4 -...- A045754(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A045754 (7-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,7*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 7^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A045754(n-k).

A113149 Row 8 of table A113143; equal to INVERT of 8-fold factorials shifted one place right.

Original entry on oeis.org

1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, 70547721068232, 5697913588192520, 505926926171909576, 48979597517592503560, 5134435963996172979912, 579379155027833982679816
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			A(x) = 1 + x + 2*x^2 + 12*x^3 + 176*x^4 + 4184*x^5 +...
= 1/(1 - x - x^2 - 9*x^3 - 153*x^4 -...- A045755(n)*x^(n+1)
-...).

Crossrefs

Cf. A113143, A045755 (8-fold factorials).

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n)); A=1/(1-x-x^2*sum(j=0,n,x^j*prod(i=0,j,8*i+1)));return(polcoeff(A,n,X))}

Formula

a(n) = Sum_{j=0..k} 8^(k-j)*A111146(k, j).
a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A045755(n-k).
Showing 1-6 of 6 results.

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