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 11 results. Next

A111556 Column 3 of triangle A111553.

Original entry on oeis.org

1, 4, 24, 184, 1664, 17024, 192384, 2366144, 31362304, 444907264, 6720628224, 107674883584, 1823884857344, 32575705493504, 612054254936064, 12071987619713024, 249477777420304384, 5392386599983366144
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

Also found in column 0 of triangle A111559, which equals the matrix inverse of A111553.

Links

Crossrefs

Cf. A111553, A111531, A111559.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+4,n+4,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))^-1)[n+4,4])}

Formula

G.f.: log(Sum_{n>=0} ((n+3)!/3!)*x^n) = Sum_{n>=1} a(n)*x^n/n.
a(n) = 4*A111531(n) for n>0.
From Groux Roland, Dec 10 2010: (Start)
6*a(n+1) = (n+5)! - 4*(n+4)! - Sum_{k=0..n-1} (n-k+3)!*a(k+1).
a(n+1) is the moment of order n for the density 6*x^3*exp(-x)/( (x^3*exp(-x)*Ei(x)-x^2-x-2)^2 + Pi^2*x^6*exp(-2*x) ) over the interval 0..infinity. (End)
a(n) = Sum_{k=0..n} A200659(n,k)*3^k. - Philippe Deléham, Nov 21 2011
G.f.: 1/(1-4x/(1-2x/(1-5x/(1-3x/(1-6x/(1-4x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011
G.f.: 1/Q(0), where Q(k) = 1 - 2*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/x - 3 - 2/(x*G(0)), where G(k) = 1 + 1/(1 - x*(k+4)/(x*(k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0), where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

A111560 Matrix logarithm of triangle A111553.

Original entry on oeis.org

0, 1, 0, 5, 2, 0, 34, 7, 3, 0, 282, 44, 10, 4, 0, 2696, 354, 60, 14, 5, 0, 28792, 3328, 470, 84, 19, 6, 0, 337072, 35144, 4344, 654, 118, 25, 7, 0, 4273632, 407984, 45320, 6008, 936, 164, 32, 8, 0, 58195072, 5137824, 521200, 62344, 8704, 1352, 224, 40, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Examples

			Triangle begins:
0;
1,0;
5,2,0;
34,7,3,0;
282,44,10,4,0;
2696,354,60,14,5,0;
28792,3328,470,84,19,6,0;
337072,35144,4344,654,118,25,7,0;
4273632,407984,45320,6008,936,164,32,8,0; ...

Crossrefs

Cf. A111553, A111557, A104986, A111541, A111549.

Programs

  • PARI
    {T(n,k)=local(M=matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))); sum(i=1,#M,(M^0-M)^i/i)[n+1,k+1]}

A111554 Column 1 of triangle A111553.

Original entry on oeis.org

1, 2, 10, 72, 632, 6352, 70912, 864192, 11371072, 160333952, 2409420032, 38428442112, 648333665792, 11538313372672, 216092576714752, 4249731095212032, 87591291183296512, 1888669450881032192
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

a(n) is the number of increasing strict binary trees with 2n-1 nodes that avoid 132. For more information about increasing strict binary trees with an associated permutation, see A245894. - Manda Riehl, Aug 07 2014

Crossrefs

Cf. A111553.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))^-1)[n+2,2])}

A111557 Column 4 of triangle A111553; also found in column 0 of triangle A111560, which equals the matrix logarithm of A111553.

Original entry on oeis.org

1, 5, 34, 282, 2696, 28792, 337072, 4273632, 58195072, 846038912, 13072140032, 213897731712, 3695682017792, 67254929193472, 1286282280266752, 25802708552696832, 541894309127053312, 11894387852938452992
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111553, A111560.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+5,n+5,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))^-1)[n+5,5])}

Formula

a(n) = A111553(n+4, 4) = A111560(n+1, 0).

A111559 Matrix inverse of triangle A111553.

Original entry on oeis.org

1, -1, 1, -4, -2, 1, -24, -4, -3, 1, -184, -24, -4, -4, 1, -1664, -184, -24, -4, -5, 1, -17024, -1664, -184, -24, -4, -6, 1, -192384, -17024, -1664, -184, -24, -4, -7, 1, -2366144, -192384, -17024, -1664, -184, -24, -4, -8, 1, -31362304, -2366144, -192384, -17024, -1664, -184, -24, -4, -9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

After initial terms, all columns are equal to -A111556.

Examples

			Triangle begins:
1;
-1,1;
-4,-2,1;
-24,-4,-3,1;
-184,-24,-4,-4,1;
-1664,-184,-24,-4,-5,1;
-17024,-1664,-184,-24,-4,-6,1;
-192384,-17024,-1664,-184,-24,-4,-7,1; ...

Crossrefs

Cf. A111553, A111556, A104984, A111540, A111548.

Programs

  • PARI
    T(n,k)=if(n

Formula

T(n, n)=1 and T(n+1, n)=-n-1, else T(n+k+1, k) = -A111556(k) for k>=1.

A111555 Column 2 of triangle A111553.

Original entry on oeis.org

1, 3, 16, 116, 1016, 10176, 113216, 1375456, 18047296, 253815936, 3805221376, 60558070016, 1019617312256, 18111737604096, 338602832961536, 6648048064792576, 136810876329865216, 2945671077411987456
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111553, A111531.

Programs

  • PARI
    {a(n)=if(n<0,0,(matrix(n+3,n+3,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))^-1)[n+3,3])}

A111558 Row sums of triangle A111553.

Original entry on oeis.org

1, 2, 9, 60, 509, 5034, 55689, 674624, 8838189, 124199454, 1861297049, 29618273604, 498716980989, 8860512190034, 165694162575369, 3254269731689064, 66994569450549869, 1443024570035017254
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111553.

Programs

  • PARI
    {a(n)=if(n<0,0,sum(k=1,n+1,(matrix(n+k,n+k,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))^-1)[n+1,k]))}

A111561 Column 1 of A111560, which is the matrix log of A111553.

Original entry on oeis.org

0, 2, 7, 44, 354, 3328, 35144, 407984, 5137824, 69566144, 1006372864, 15481560064, 252326173824, 4344015683584, 78793242566144, 1502374856981504, 30052439647908864, 629485600310349824, 13783057303819485184
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111560, A111553.

Programs

  • PARI
    {a(n)=local(M=matrix(n+2,n+2,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+2,2])}

A111562 Column 2 of A111560, which is the matrix log of A111553.

Original entry on oeis.org

0, 3, 10, 60, 470, 4344, 45320, 521200, 6513280, 87613440, 1260188800, 19286781440, 312884243840, 5363632995840, 96904980170240, 1840977689943040, 36700487712701440, 766296476640215040, 16728728381231472640
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111560, A111553.

Programs

  • PARI
    {a(n)=local(M=matrix(n+3,n+3,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+3)!/3!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+3,3])}

A111544 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+3 of T), or [T^p](m,0) = p*T(p+m,p+3) for all m>=1 and p>=-3.

Original entry on oeis.org

1, 1, 1, 5, 2, 1, 33, 9, 3, 1, 261, 57, 15, 4, 1, 2361, 441, 99, 23, 5, 1, 23805, 3933, 783, 165, 33, 6, 1, 263313, 39249, 7083, 1383, 261, 45, 7, 1, 3161781, 430677, 71415, 13083, 2361, 393, 59, 8, 1, 40907241, 5137641, 789939, 136863, 23805, 3861, 567, 75, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

Column 0 equals A111530 (related to log of factorial series). Column 3 (A111547) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111549.

Examples

			SHIFT_LEFT(column 0 of T^-3) = -3*(column 0 of T);
SHIFT_LEFT(column 0 of T^-2) = -2*(column 1 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 2 of T);
SHIFT_LEFT(column 0 of log(T)) = column 3 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
5,2,1;
33,9,3,1;
261,57,15,4,1;
2361,441,99,23,5,1;
23805,3933,783,165,33,6,1;
263313,39249,7083,1383,261,45,7,1;
3161781,430677,71415,13083,2361,393,59,8,1; ...
After initial term, column 2 is 3 times column 0.
Matrix inverse T^-1 = A111548 starts:
1;
-1,1;
-3,-2,1;
-15,-3,-3,1;
-99,-15,-3,-4,1;
-783,-99,-15,-3,-5,1;
-7083,-783,-99,-15,-3,-6,1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 2 of T.
Matrix logarithm log(T) = A111549 is:
0;
1,0;
4,2,0;
23,6,3,0;
165,32,9,4,0;
1383,222,47,13,5,0;
13083,1824,321,70,18,6,0; ...
compare column 0 of log(T) to column 3 of T.

Links

Crossrefs

Cf. A111545 (column 1), A111546 (column 2), A111547 (column 3), A111552 (row sums), A111548 (matrix inverse), A111549 (matrix log); related tables: A111528, A104980, A111536, A111553.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = If[nJean-François Alcover, Aug 09 2018, from PARI *)
  • PARI
    T(n,k)=if(n

Formula

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+2, 2)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+3, 2) = 3*T(n+1, 0), T(n+4, 4) = T(n+1, 0), for n>=0.
Showing 1-10 of 11 results. Next

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

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