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 1-9 of 9 results.

A111537 Column 1 of triangle A111536.

Original entry on oeis.org

1, 2, 8, 44, 296, 2312, 20384, 199376, 2138336, 24936416, 314142848, 4252773824, 61594847360, 950757812864, 15586971531776, 270569513970944, 4959071121374720, 95721139472072192, 1941212789888952320, 41271304403571227648, 918030912312297752576, 21325054720042613565440
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

Row sums of triangle in A200659. - Philippe Deléham, Nov 21 2011

References

  • A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p.141 (10.19).
  • H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.

Links

Crossrefs

Cf. A111536, A111529, A200659.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
      n*(n+1)! -add((n-k+1)!*a(k), k=1..n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 06 2013
  • Mathematica
    a[n_] := a[n] = If[n==0, 1, n*(n+1)! - Sum[(n-k+1)!*a[k], {k, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 13 2017, after Alois P. Heinz *)
  • 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+1)!/1!*x^i)),m-j-1))))^-1)[n+2,2])}

Formula

a(n) = A111536(n+1, 1) = 2*A111536(n, 0) = 2*A111529(n) for n >= 1.
G.f.: log(Sum_{n>=0} (n+1)!*x^n) = Sum_{n>=1} a(n)*x^n/n.
a(n+1) = (n+3)! - 2*(n+2)! - Sum_{k=0..n-1} (n-k+1)!*a(k+1).
a(n+1) is the moment of order n for the measure of density x*exp(-x)/((x*exp(-x)*Ei(x)-1)^2+(Pi*x*exp(-x))^2) on the interval 0..infinity.
G.f.: 1/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-5*x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011
G.f. (1-U(0))/x; where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jun 29 2012
G.f. -1 + 1/x + U(0)/x where U(k) = 2*x - 1 + 2*x*k - x^2*(k+1)*(k+2)/U(k+1), U(0)=x - W(1,1;-x)/W(1,2;-x), W(a,b,x)= 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]; (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 15 2012
G.f.: 1/Q(0), where Q(k) = 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/x - 1/( x*G(0)), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 03 2013
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020

A111541 Matrix logarithm of triangle A111536.

Original entry on oeis.org

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, 60062352, 7155104
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

Column 0 (A111538) is found in column 2 of triangle A111536.

Examples

			Triangle begins:
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; ...

Crossrefs

Cf. A111536, A111538 (column 0), A111542 (column 1), A111543 (column 2).

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+1)!/1!*x^i)),m-j-1))))); return(if(n

A111538 Column 2 of triangle A111536; also equals column 0 of triangle A111541, which is the matrix log of triangle A111536.

Original entry on oeis.org

1, 3, 14, 84, 600, 4908, 44952, 454344, 5016768, 60062352, 775089312, 10728930912, 158638465536, 2496437377728, 41674737264000, 735831528563328, 13704461848753152, 268562085051533568, 5525005742876244480
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Crossrefs

Cf. A111536, A111541.

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+1)!/1!*x^i)),m-j-1))))^-1)[n+3,3])}

Formula

a(n) = A111536(n+2, 2) = A111541(n+1, 0).

A111540 Matrix inverse of triangle A111536.

Original entry on oeis.org

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, -2138336, -199376, -20384, -2312, -296, -44, -8, -2, -9, 1, -24936416, -2138336, -199376, -20384, -2312, -296
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

The column sequences are derived from the logarithm of a factorial series (cf. A111537).

Examples

			Triangle begins:
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; ...
After initial terms, all columns are equal to -A111537.

Crossrefs

Cf. A111536, A111537.

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) = -A111537(k) for k>=1.

A111542 Column 1 of triangle A111541, which is the matrix logarithm of A111536.

Original entry on oeis.org

0, 2, 5, 22, 128, 896, 7220, 65336, 653720, 7155104, 84998768, 1089232160, 14981704736, 220233312896, 3447195190592, 57261708795776, 1006401042534272, 18663532970127872, 364283224523605760, 7466218532765196800
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Comments

Column 0 of triangle A111541 is A111536.

Crossrefs

Cf. A111541 (triangle), A111538 (column 0), A111543 (column 2), A111536.

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

A111543 Column 2 of triangle A111541, which is the matrix logarithm of A111536.

Original entry on oeis.org

0, 3, 8, 36, 212, 1496, 12128, 110288, 1108064, 12171872, 145061120, 1864321472, 25710635648, 378871778432, 5943632568320, 98936446059776, 1742232571097600, 32367994818881024, 632845309575139328, 12991224275641441280
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Crossrefs

Cf. A111541 (triangle), A111538 (column 0), A111542 (column 1), A111536.

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

A111539 Row sums of triangle A111536.

Original entry on oeis.org

1, 2, 7, 34, 211, 1564, 13291, 126430, 1326211, 15184744, 188348659, 2515991290, 36018842875, 550317892372, 8940695112331, 153950966471734, 2801321782936819, 53720913548093344, 1083061044893428675
Offset: 0

Views

Author

Paul D. Hanna, Aug 06 2005

Keywords

Crossrefs

Cf. A111536.

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

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.

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 1-9 of 9 results.

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

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