A111546 -id:A111546 - cp's OEIS frontend

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) = pT(p+m,p+3) for all m>=1 and p>=-3.

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

Paul D. Hanna, Aug 07 2005

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.

			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.

Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

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.

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

PARI
```
T(n,k)=if(n
				
```

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.

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

Paul D. Hanna, Aug 07 2005

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

			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; ...

Cf. A111544, A111546, A111541.

{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]}

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

Paul D. Hanna, Aug 07 2005

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

			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; ...

Cf. A111544, A111546, A111540.

PARI
```
T(n,k)=if(n
				
```

T(n, n)=1 and T(n+1, n)=-n-1, else T(n+k+1, k) = -A111546(k) for k>=1.

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A200659 Triangle T(n,k), read by rows, given by (1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 13, 21, 9, 1, 71, 132, 76, 16, 1, 461, 955, 670, 200, 25, 1, 3447, 7782, 6309, 2374, 435, 36, 1, 29093, 70441, 63833, 28413, 6713, 833, 49, 1, 273343, 701352, 694500, 351512, 99868, 16240, 1456, 64, 1

Offset: 0

Philippe Deléham, Nov 20 2011

			Triangle begins :
1
1, 1
3, 4, 1
13, 21, 9, 1
71, 132, 76, 16, 1
461, 955, 670, 200, 25, 1
3447, 7782, 6309, 2374, 435, 36, 1
29093, 70441, 63833, 28413, 6713, 833, 49, 1
273343, 701352, 694500, 351512, 99868, 16240, 1456, 64, 1

Cf. A111537, A111546, A111556, A167872

Sum_{k, 0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A003319(n), A111537(n), A111546(n), A111556(n), A177354(n-1) for x = -2,-1,0,1,2,3,4 respectively.
Sum_ {k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A111537(n), A167872(n) for x = 0,1,2 respectively.
T(k+1,k)=(k+1)^2.

A177354 a(n) is the moment of order n for the density measure 24x^4exp(-x)/( (x^4exp(-x)Ei(x) - x^3 - x^2 - 2x - 6)^2 + Pi^2x^8exp(-2x) ) over the interval 0..infinity.

Original entry on oeis.org

5, 35, 305, 3095, 35225, 439775, 5939225, 85961375, 1324702025, 21632195375, 372965377625, 6769644905375, 129049505347625, 2578419996023375, 53898389265685625, 1176832196718869375, 26798832693476455625, 635575680349115699375, 15677971277701873945625, 401729457433222058609375

Offset: 0

Groux Roland, Dec 10 2010

nonn

Ei(.) is the exponential integral.
This is the case k=4 in the family a(n,k) = (1/k!)*( (n+k+2)!-(k+1)*(n+k+1)! -Sum_{i=0..n-1} (n+k-i)!*a(i,k) ). The values k = 0 to 3 are represented by A003319, A111537, A111546, and A111556.
a(n,k) is the moment of order n for the density k!*x^k*exp(-x)/((x^k*exp(-x)*Ei(x) - Pk(x))^2 + Pi^2*x^(2*k)*exp(-2*x)) on the interval 0..infinity with polynomials Pk(x) = Sum_{i=0..k-1} (k-1-i)!*x^i.

R. Groux, Polynômes orthogonaux et transformations intégrales, Cépadués, 2008, 125-129.

a(n)=if(n==0, 5, (1/24)*( (n+6)! -5*(n+5)! -sum(i=0,n-1, (n+4-i)!*a(i) ) ) ); \\ Joerg Arndt, May 04 2013

a(n) = (1/24)*( (n+6)! - 5*(n+5)! - Sum_{i=0..n-1} (n+4-i)!*a(i) ).
a(n) = 5*A111532(n+1) (conjecture). - R. J. Mathar, Dec 14 2010
G.f.: 1/x/Q(0) - 1/x, where Q(k) = 1 - 3*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
G.f.: (1-x-2/G(0))/x^2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: 1/x^2 - 5/x - 2/(x^2*G(0)), where G(k) = 1 + 1/(1 - x*(k+5)/(x*(k+5) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

A230253 Coefficients of power series A(x) such that coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.

Original entry on oeis.org

1, 3, 3, 6, 27, 162, 1206, 10476, 103059, 1125738, 13473378, 174997908, 2448791838, 36706645908, 586646510796, 9957100024152, 178868488496643, 3390603439026618, 67639341903290730, 1416612563019545220, 31079692422132040170, 712855563504590236860, 17061654943814209044660

Offset: 0

Paul D. Hanna, Oct 13 2013

nonn

			G.f. A(x) = 1 + 3*x + 3*x^2 + 6*x^3 + 27*x^4 + 162*x^5 + 1206*x^6 +...
The coefficients in A(x)^n begin:
n=1: [(1),3,   3,    6,    27,    162,   1206,   10476,   103059, ...];
n=2: [1, (6), 15,   30,    99,    522,   3582,   29484,   278883, ...];
n=3: [1,  9, (36),  99,   297,   1323,   8208,   63342,   572751, ...];
n=4: [1, 12,  66, (240),  783,   3132,  17298,  123552,  1060155, ...];
n=5: [1, 15, 105,  480, (1800),  7083,  35415,  231660,  1869885, ...];
n=6: [1, 18, 153,  846,  3672, (15120), 71415,  428490,  3226797, ...];
n=7: [1, 21, 210, 1365,  6804,  30240,(141120), 789939,  5529573, ...];
n=8: [1, 24, 276, 2064, 11682,  56736, 270720,(1451520), 9485343, ...];
n=9: [1, 27, 351, 2970, 18873, 100440, 500904, 2643840,(16329600), ...]; ...
where the coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.
Notice that the diagonal above the main diagonal forms A111546.

Cf. A075834, A090753, A111546.

{a(n)=polcoeff(x/serreverse(sum(m=1, n+1, (m+1)!/2*x^m)+x^2*O(x^n)), n)}

[x^n] A(x)^n = A111546(n) for n>=0.

cp's OEIS Frontend

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.

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A111549 Matrix logarithm of triangle A111544.

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A111548 Matrix inverse of triangle A111544.

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A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

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PARI

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A200659 Triangle T(n,k), read by rows, given by (1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

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A177354 a(n) is the moment of order n for the density measure 24*x^4*exp(-x)/( (x^4*exp(-x)*Ei(x) - x^3 - x^2 - 2*x - 6)^2 + Pi^2*x^8*exp(-2*x) ) over the interval 0..infinity.

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A230253 Coefficients of power series A(x) such that coefficient of x^n in A(x)^(n+1) equals (n+1)*(n+2)!/2 for n>=0.

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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) = pT(p+m,p+3) for all m>=1 and p>=-3.

A177354 a(n) is the moment of order n for the density measure 24x^4exp(-x)/( (x^4exp(-x)Ei(x) - x^3 - x^2 - 2x - 6)^2 + Pi^2x^8exp(-2x) ) over the interval 0..infinity.