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

A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 26, 54, 32, 1, 1, 2, 8, 41, 158, 235, 64, 1, 1, 2, 9, 60, 364, 1282, 1237, 128, 1, 1, 2, 10, 83, 708, 4409, 13158, 7790, 256, 1, 1, 2, 11, 110, 1226, 11428, 67563, 163354
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 30 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 the row sums of R(m,n,k) are given by row m; example: m = 1, R(1,n,k) = A084938(n,k) and A051295 gives the row sums of A084938.
Square array of INVERT of m-fold factorials.

Examples

			Table begins:
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...
1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, ...
1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
1, 1, 2, 7, 41, 364, 4409, 67573, 1248626, 26948347, 664414997, ...
1, 1, 2, 8, 60, 708, 11428, 232756, 5704964, 163192820, 5331728964, ...
1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, ...
1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, ...
1, 1, 2, 11, 141, 2928, 82597, 2925973, 124502114, 6179425823, 350316271761, ...
1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, ...
1, 1, 2, 13, 215, 5758, 208643, 9481141, 517310894, 32922122485, 2393313188039, ...
1, 1, 2, 14, 258, 7686, 309322, 15604654, 945111938, 66766075046, 5387893860042, ...

Crossrefs

Cf. A051295 (row n=1), A112934 (row n=2), A113144 (row n = 3), A113145 (row n=4), A113146 (row n=5), A113147 (row n = 6), A113148 (row n=7), A113149 (row n=8).

Programs

  • PARI
    {T(n,k)=local(x=X+X*O(X^k),y=Y+Y*O(Y^k));A=1/(1-x*y*sum(j=0,k,x^j*prod(i= 0,j-1,y+i)));return(sum(m=0,k,n^(k-m)*polcoeff(polcoeff(A,k,X),m,Y)))}

Formula

T(n, k) = Sum_{j=0..k} n^(k-j)*A111146(k, j).

A113326 Table T(n,k), n>=1 and k>=0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 8, 5, 1, 4, 18, 36, 15, 1, 5, 32, 117, 176, 54, 1, 6, 50, 272, 801, 928, 235, 1, 7, 72, 525, 2400, 5724, 5296, 1237, 1, 8, 98, 900, 5675, 21792, 42633, 33024, 7790, 1, 9, 128, 1421, 11520, 62650, 203008, 331911, 227776, 57581
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			Table begins:
1,1,2,5,15,54,235,1237,7790,57581, 489231, ...
1,2,8,36,176,928,5296,33024,227776,1757504, ...
1,3,18,117,801,5724,42633,331911,2717874,23620329, ...
1,4,32,272,2400,21792,203008,1940224,19065344,193410560, ...
1,5,50,525,5675,62650,703975,8042625,93454750,1106250125, ...
1,6,72,900,11520,149904,1976400,26363232,355648320, ...
1,7,98,1421,21021,315168,4774021,72945859,1123559906, ...
1,8,128,2112,35456,601984,10306048,177639936,3080264704, ...
1,9,162,2997,56295,1067742,20392803,391614669,7555447854, ...
1,10,200,4100,85200,1785600,37644400,797224000,16946456000,
...

Crossrefs

Cf. A111146, A051295 (row n=1), A113327 (row n=2), A113328 (row n=3), A113329 (row n=4), A113330 (row n=5), A113331 (row n=6).

Programs

  • PARI
    {T(n,k)=local(y=Y+Y*O(Y^k)); polcoeff(1/(1-n/(n-1)!*y*sum(j=0,k,(n-1+j)!*y^j)),k,Y)}

Formula

T(n, k) = Sum_{j=0..k} n^j*A111146(k, j).
G.f. for row n: Sum_{k>=0}T(n, k)*y^k = 1/(1-n/(n-1)!*y*Sum_{j>=0}(n-1+j)!*y^j), for n>=1.

A113327 a(n) = Sum_{k=0..n} 2^k*A111146(n,k).

Original entry on oeis.org

1, 2, 8, 36, 176, 928, 5296, 33024, 227776, 1757504, 15269888, 149327616, 1632715520, 19758502912, 261836047360, 3763432774656, 58208166178816, 962637398577152, 16934963591229440, 315578267054112768
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 2*x + 8*x^2 + 36*x^3 + 176*x^4 + 928*x^5 +..) =
1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 + 5!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=2,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - 2*x*Sum_{k>=0} (k+1)!*x^k ).

A113328 a(n) = Sum_{k=0..n} 3^k*A111146(n,k).

Original entry on oeis.org

1, 3, 18, 117, 801, 5724, 42633, 331911, 2717874, 23620329, 220260789, 2228505372, 24681015981, 300506801715, 4017984855786, 58675338993069, 928673101727001, 15804592586240220, 287174716511520033, 5538727108037507535
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 3*x + 18*x^2 + 117*x^3 + 801*x^4 + 5724*x^5 +..)
= 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 + 6!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=3,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (3/2)*x*Sum_{k>=0} (k+2)!*x^k ).

A113329 a(n) = Sum_{k=0..n} 4^k*A111146(n,k).

Original entry on oeis.org

1, 4, 32, 272, 2400, 21792, 203008, 1940224, 19065344, 193410560, 2038078464, 22490167296, 262429339648, 3271314362368, 43955391856640, 640254018879488, 10121874150653952, 173145693892509696, 3186234896556752896
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 4*x + 32*x^2 + 272*x^3 + 2400*x^4 + 21792*x^5 +..)
= 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 + 7!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113328 (y=3), A113330 (y=5), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=4,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (2/3)*x*Sum_{k>=0} (k+3)!*x^k ).

A113330 a(n) = Sum_{k=0..n} 5^k*A111146(n,k).

Original entry on oeis.org

1, 5, 50, 525, 5675, 62650, 703975, 8042625, 93454750, 1106250125, 13377432875, 165950540250, 2124087269375, 28260204825625, 394301229688750, 5824314672613125, 91872380184761875, 1557002324898406250
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 5*x + 50*x^2 + 525*x^3 + 5675*x^4 + 62650*x^5 +..)
= 1/(1 - 5/4!*x*(4! + 5!*x + 6!*x^2 + 7!*x^3 + 8!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113331 (y=6).

Programs

  • PARI
    {a(n)=local(y=5,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (5/24)*x*Sum_{k>=0} (k+4)!*x^k ).

A113331 a(n) = Sum_{k=0..n} 6^k*A111146(n,k).

Original entry on oeis.org

1, 6, 72, 900, 11520, 149904, 1976400, 26363232, 355648320, 4854292416, 67114780416, 941774874624, 13451571452160, 196362144456192, 2945496714485760, 45717104468689920, 740282299231703040
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 26 2005

Keywords

Examples

			A(x) = (1 + 6*x + 72*x^2 + 900*x^3 + 11520*x^4 + 149904*x^5 +..)
= 1/(1 - 6/5!*x*(5! + 6!*x + 7!*x^2 + 8!*x^3 + 9!*x^4 +..) ).

Crossrefs

Cf. A111146, A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5).

Programs

  • PARI
    {a(n)=local(y=6,x=X+X*O(X^n)); polcoeff(1/(1 - y/(y-1)!*x*sum(k=0,n,(y-1+k)!*x^k)),n,X)}

Formula

G.f.: A(x) = 1/(1 - (1/20)*x*Sum_{k>=0} (k+5)!*x^k ).

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).
Showing 1-10 of 13 results. Next

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