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 1-3 of 3 results.

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.

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

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.
Showing 1-3 of 3 results.

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