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 10 results.

A111546 Column 2 of triangle A111544.

Original entry on oeis.org

1, 3, 15, 99, 783, 7083, 71415, 789939, 9485343, 122721723, 1701224775, 25156450179, 395362560303, 6583219735563, 115817825451735, 2147443419579219, 41868118883289663, 856527397513863003, 18350158259899381095, 410942059850878349859, 9603217302778609785423
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

Also forms the columns of triangle A111548, which is the matrix inverse of triangle A111544.

Links

Crossrefs

Cf. A111544, A111530, A111548, A230253.
Cf. A005412.

Programs

  • Haskell
    a111546 n = a111546_list !! n
a111546_list = 1 : f 2 [1] where
   f v ws@(w:_) = y : f (v + 1) (y : ws) where
                  y = v * w + (sum $ zipWith (*) ws $ reverse ws)
-- Reinhard Zumkeller, Jan 24 2014
  • 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+2)!/2!*x^i)),m-j-1))))^-1)[n+3,3])}
  • PARI
    a(n)=(1/2)*((n+3)!-3*(n+2)!-sum(k=0,n-2,(n+1-k)!*a(k+1))) \\ Formula by Groux Roland, implemented & checked to conform to given terms by M. F. Hasler, Dec 12 2010

Formula

G.f.: log(Sum_{n>=0} (n+2)!/2!*x^n) = Sum_{n>=1} a(n)*x^n/n. a(n) = 3*A111530(n) = -A111548(n+1, 0) for n>0.
a(n+1) = (1/2)*((n+4)!-3*(n+3)!-Sum_{k=0..n-1} (n+2-k)!*a(k+1)).
a(n+1) is the moment of order n for the measure of density: 2*x^2*exp(-x)/((x^2*exp(-x)*Ei(x)-x-1)^2+Pi^2*x^4*exp(-2*x)), on the interval 0..infinity. - Groux Roland, Dec 10 2010
a(n) = Sum_{k=0..n} A200659(n,k)*2^k. - Philippe Deléham, Nov 21 2011
G.f.: 1/(1-3x/(1-2x/(1-4x/(1-3x/(1-5x/(1-4x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011
G.f. 2 - U(0) where U(k)= 1 - x*(k+1)/(1 - x*(k+3)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jun 29 2012
G.f. -1/G(0) where G(k) = x - 1 - k*x - x*(k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 13 2012
G.f.: A(x) = 1/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
G.f.: 1/Q(0), where Q(k)= 1 - x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/x -2 -2/(x*G(0)), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: 1/x - 2 - 1/(x*W(0)), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
G.f.: W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^3 / 2. - Vaclav Kotesovec, May 24 2025

A111549 Matrix logarithm of triangle A111544.

Original entry on oeis.org

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, 136863, 17016, 2607, 486, 103, 24, 7, 0, 1562715, 176112, 24099, 3990, 747, 148, 31, 8, 0, 19301319, 1993392, 247527, 37182, 6351, 1140, 207, 39, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

Column 0 (A111546) is found in column 3 of triangle A111544.

Examples

			Triangle begins:
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;
136863,17016,2607,486,103,24,7,0;
1562715,176112,24099,3990,747,148,31,8,0; ...

Crossrefs

Cf. A111544, A111546, A111541.

Programs

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

Formula

T(n+1, 0) = A111546(n) = A111544(n+3, 3).

A111548 Matrix inverse of triangle A111544.

Original entry on oeis.org

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, -71415, -7083, -783, -99, -15, -3, -7, 1, -789939, -71415, -7083, -783, -99, -15, -3, -8, 1, -9485343, -789939, -71415, -7083, -783, -99, -15, -3, -9, 1, -122721723, -9485343, -789939, -71415
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

The column sequences are all equal after the initial terms and are derived from the logarithm of a factorial series (cf. A111546).

Examples

			Triangle begins:
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;
-71415,-7083,-783,-99,-15,-3,-7,1;
-789939,-71415,-7083,-783,-99,-15,-3,-8,1; ...

Crossrefs

Cf. A111544, A111546, A111540.

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) = -A111546(k) for k>=1.

A111545 Column 1 of triangle A111544.

Original entry on oeis.org

1, 2, 9, 57, 441, 3933, 39249, 430677, 5137641, 66103533, 911973249, 13428837477, 210259505241, 3489291147933, 61200522593649, 1131630622515477, 22007781151434441, 449197758198576333, 9603342828706622049
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111544.

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+2)!/2!*x^i)),m-j-1))))^-1)[n+2,2])}

A111547 Column 3 of triangle A111544; also found in column 0 of triangle A111549, which equals the matrix logarithm of A111544.

Original entry on oeis.org

1, 4, 23, 165, 1383, 13083, 136863, 1562715, 19301319, 256191363, 3636036783, 54956529675, 881578601559, 14964805041363, 268105552191423, 5057384615702235, 100224731537223399, 2082402995330965923
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111544, A111549.

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+2)!/2!*x^i)),m-j-1))))^-1)[n+4,4])}

Formula

a(n) = A111544(n+3, 3) = A111549(n+1, 0).

A111552 Row sums of triangle A111544.

Original entry on oeis.org

1, 2, 8, 46, 338, 2930, 28726, 311342, 3679778, 47000002, 644538614, 9443216366, 147222001954, 2434125748082, 42555399987686, 784634319490894, 15221022251713538, 309977607819594818, 6613678784212854646
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111544.

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+2)!/2!*x^i)),m-j-1))))^-1)[n+1,k]))}

A111550 Column 1 of A111549, which is the matrix log of A111544.

Original entry on oeis.org

0, 2, 6, 32, 222, 1824, 17016, 176112, 1993392, 24438960, 322294896, 4548010032, 68385367152, 1091838106800, 18454096189296, 329306074785072, 6189015238217712, 122232512688657840, 2531600753529542256
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111549, A111544.

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+2)!/2!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+2,2])}

A111551 Column 2 of A111549, which is the matrix log of A111544.

Original entry on oeis.org

0, 3, 9, 47, 321, 2607, 24099, 247527, 2783331, 33924303, 445016619, 6249234807, 93541817331, 1487200667103, 25037315924859, 445123900236807, 8336458657796931, 164100631571947503, 3388128151043405259
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Crossrefs

Cf. A111549, A111544.

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+2)!/2!*x^i)),m-j-1))))); if(n<0,0,sum(i=1,#M,(M^0-M)^i/i)[n+3,3])}

A111553 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+4 of T), or [T^p](m,0) = p*T(p+m,p+4) for all m>=1 and p>=-4.

Original entry on oeis.org

1, 1, 1, 6, 2, 1, 46, 10, 3, 1, 416, 72, 16, 4, 1, 4256, 632, 116, 24, 5, 1, 48096, 6352, 1016, 184, 34, 6, 1, 591536, 70912, 10176, 1664, 282, 46, 7, 1, 7840576, 864192, 113216, 17024, 2696, 416, 60, 8, 1, 111226816, 11371072, 1375456, 192384, 28792, 4256, 592, 76, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2005

Keywords

Comments

Column 0 equals A111531 (related to log of factorial series). Column 4 (A111557) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111560.

Examples

			SHIFT_LEFT(column 0 of T^-4) = -4*(column 0 of T);
SHIFT_LEFT(column 0 of T^-3) = -3*(column 1 of T);
SHIFT_LEFT(column 0 of T^-2) = -2*(column 2 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 3 of T);
SHIFT_LEFT(column 0 of log(T)) = column 4 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 5 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1,1;
6,2,1;
46,10,3,1;
416,72,16,4,1;
4256,632,116,24,5,1;
48096,6352,1016,184,34,6,1;
591536,70912,10176,1664,282,46,7,1;
7840576,864192,113216,17024,2696,416,60,8,1; ...
After initial term, column 3 is 4 times column 0.
Matrix inverse T^-1 = A111559 starts:
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; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 3 of T.
Matrix logarithm log(T) = A111560 is:
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; ...
compare column 0 of log(T) to column 4 of T.

Links

Crossrefs

Cf. A111531 (column 0), A111554 (column 1), A111555 (column 2), A111556 (column 3), A111557 (column 4), A111558 (row sums), A111559 (matrix inverse), A111560 (matrix log); related tables: A111528, A104980, A111536, A111544.

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+3, 3)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+4, 3) = 4*T(n+1, 0), T(n+5, 5) = T(n+1, 0), for n>=0.

A111536 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+2 of T), or [T^p](m,0) = p*T(p+m,p+2) for all m>=1 and p>=-2.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 22, 8, 3, 1, 148, 44, 14, 4, 1, 1156, 296, 84, 22, 5, 1, 10192, 2312, 600, 148, 32, 6, 1, 99688, 20384, 4908, 1156, 242, 44, 7, 1, 1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1, 12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

Column 0 equals A111529 (related to log of factorial series).
Column 2 (A111538) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111541.

Examples

			SHIFT_LEFT(column 0 of T^-2) = -2*(column 0 of T);
SHIFT_LEFT(column 0 of T^-1) = -1*(column 1 of T);
SHIFT_LEFT(column 0 of log(T)) = column 2 of T;
SHIFT_LEFT(column 0 of T^1) = 1*(column 3 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 4 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
1;
1, 1;
4, 2, 1;
22, 8, 3, 1;
148, 44, 14, 4, 1;
1156, 296, 84, 22, 5, 1;
10192, 2312, 600, 148, 32, 6, 1;
99688, 20384, 4908, 1156, 242, 44, 7, 1;
1069168, 199376, 44952, 10192, 2084, 372, 58, 8, 1;
12468208, 2138336, 454344, 99688, 20012, 3528, 544, 74, 9, 1; ...
...
After initial term, column 1 is twice column 0.
Matrix inverse T^-1 = A111540 starts:
1;
-1, 1;
-2, -2, 1;
-8, -2, -3, 1;
-44, -8, -2, -4, 1;
-296, -44, -8, -2, -5, 1;
-2312, -296, -44, -8, -2, -6, 1;
-20384, -2312, -296, -44, -8, -2, -7, 1;
-199376, -20384, -2312, -296, -44, -8, -2, -8, 1; ...
where columns are all equal after initial terms;
compare columns of T^-1 to column 1 of T.
Matrix logarithm log(T) = A111541 is:
0;
1, 0;
3, 2, 0;
14, 5, 3, 0;
84, 22, 8, 4, 0;
600, 128, 36, 12, 5, 0;
4908, 896, 212, 58, 17, 6, 0;
44952, 7220, 1496, 360, 90, 23, 7, 0;
454344, 65336, 12128, 2652, 602, 134, 30, 8, 0;
5016768, 653720, 110288, 22320, 4736, 974, 192, 38, 9, 0; ...
compare column 0 of log(T) to column 2 of T.

Links

Crossrefs

Cf. A111537 (column 1), A111538 (column 2), A111539 (row sums), A111540 (matrix inverse), A111541 (matrix log); related tables: A111528, A104980, A111544, A111553.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = If[nJean-François Alcover, Jan 24 2017, adapted 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+1, 1)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+2, 1) = 2*T(n+1, 0), T(n+3, 3) = T(n+1, 0), for n>=0.
Showing 1-10 of 10 results.

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

Content is available under The OEIS End-User License Agreement.