A089505 -id:A089505 - cp's OEIS frontend

A089506 Denominators used in A089505 to compute column sequences of triangle A089504.

1, 3, 27, 513, 540702, 8891844390, 27306854121690, 94235953573952190, 1684561906087969348440, 1106757172299795861925080, 8644064410182787098480243878401800, 289900692457541891469406183557646377576408600, 1206267931807293851480420134690207296382114287151076400

Offset: 1

Wolfdieter Lang, Dec 01 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..50

a[n_, m_] := (-1)^(n-m)*FactorialPower[m+2, 3]^(n-1)/(Product[ FactorialPower[m+2, 3] - FactorialPower[r+2, 3], {r, 1, m-1}]*Product[ FactorialPower[r+2, 3] - FactorialPower[m+2, 3], {r, m+1, n}]); a[n_] := LCM @@ Table[Denominator[a[n, m]], {m, 1, n}]; Array[a, 11] (* Jean-François Alcover, Sep 02 2016 *)

a(n) = lcm(seq(denominator(a(n, m)), m=1..n)) with the a(n, m) formula given in A089505(n, m) but without the D(n) factor in front and lcm denotes the least common multiple of a set of numbers.

a(12)-a(13) from Vincenzo Librandi, Mar 15 2018

A089504 A generalization of triangle A071951 (Legendre-Stirling).

Original entry on oeis.org

1, 6, 1, 36, 30, 1, 216, 756, 90, 1, 1296, 18360, 6156, 210, 1, 7776, 441936, 387720, 31356, 420, 1, 46656, 10614240, 23705136, 4150440, 119556, 756, 1, 279936, 254788416, 1432922400, 521757936, 29257200, 373572, 1260, 1, 1679616

Offset: 1

Wolfdieter Lang, Dec 01 2003

This triangle underlies the array entry A078741 ((3,3)-generalized Stirling2).

For the computation of the column sequences see A089505.

			[1]; [6,1]; [36,30,1]; [216,756,90,1]; ...
a(3,2) = 30 = ((-1)*(3*2*1)^1 + 4*(4*3*2)^1)/3.

R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013.
R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.
W. Lang, First 8 rows.

Cf. A071951 (Legendre-Stirling, (2, 2) case).
The column sequences (without leading zeros) are A000400 (powers of 6), A089507, A089513-4, etc.
Cf. A008279, A089505, A089506.

max = 10; f[m_] := 1/Product[1 - FactorialPower[r + 2, 3]*x, {r, 1, m}]; col[m_] := CoefficientList[f[m] + O[x]^(max - m + 1), x]; a[n_, m_] := col[m][[n - m + 1]]; Table[a[n, m], {n, 1, max}, {m, 1, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

G.f. for m-th column sequence (without leading zeros and m>=1) is 1/Product_{r=1..m} 1-fallfac(r+2, 3)*x with fallfac(n, k) := A008279(n, k) (falling factorials).

a(n, m) = Sum_{p=1..m} A089505(m, p)*((p+2)*(p+1)*p)^(n-m))/D(m) if n>=m>=1 else 0; with D(m) := A089506(m).

A089513 Third column of triangle A089504.

Original entry on oeis.org

1, 90, 6156, 387720, 23705136, 1432922400, 86230132416, 5179923146880, 310942155338496, 18660051727004160, 1119687641441381376, 67183287394552842240, 4031045937469026349056, 241863924899255181189120

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A089507 with powers of 60.

Harvey P. Dale, Table of n, a(n) for n = 0..562
Index entries for linear recurrences with constant coefficients, signature (90,-1944,8640).

Cf. A089505, A089507, A089514.

CoefficientList[Series[1/(1-90 x+1944 x^2-8640 x^3),{x,0,20}],x] (* or *) LinearRecurrence[{90,-1944,8640},{1,90,6156},20] (* Harvey P. Dale, Jul 07 2021 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)).
a(n)=A089504(n+3, 3), n>=0.
a(n)= (50*(5*4*3)^n - 24*(4*3*2)^n + (3*2*1)^n)/27 = (6^n)*(5*10^(n+1) - 3*2^(2*n+3) + 1)/27.

A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, -1, 4, 1, -8, 10, -1, 12, -30, 20, 1, -64, 600, -1600, 1225, -1, 80, -1000, 4000, -6125, 3136, 1, -96, 1500, -8000, 18375, -18816, 7056, -1, 448, -21000, 280000, -1500625, 3687936, -4148928, 1728000, 1, -512, 28000, -448000, 3001250, -9834496, 16595712, -13824000, 4492125, -1

Offset: 3

Wolfdieter Lang, Dec 01 2003

The formula for the column no. k sequence of array A078741 is c(k;n) = b(k-2)*sum(a(k,m)*fallfac(m+2,3)^n,m=1..k-2),n>=0, k>=3 and fallfac(p,3) and b(n) are defined in the formula below.

			The third (k=5) column sequence of array A078741 is A078741(n+3,5)=c(5; n)= b(3)*(1*(3*2*1)^n -8*(4*3*2)^n +10*(5*4*3)^n), with b(3)= N(3)/A090220(3)=3/1=3, n>=0. This is 9*A089518.
The fifth (k=7) column sequence of array A078741 is A078741(n+3,7)=c(7; n)= b(5)*(1*(3*2*1)^n -64*(4*3*2)^n +600*(5*4*3)^n -1600*(6*5*4)^n +1225*(7*6*5)^n), with b(5)= N(5)/A090220(5)=3/2, n>=0. This is the sequence [243, 149580, 49658508, 13062960720,... ] which has a factor of 27.
Triangle begins:
  [1];
  [-1,4];
  [1,-8,10];
  [-1,12,-30,20];
  [1,-64,600,-1600,1225];
  ...

Wolfdieter Lang, First 10 rows.

Companion sequence A090220 for denominators D(m).

a(n, m) = A089505(n-2, m)*(sum(A089517(n, p)/fallfac(m+2, 3)^p, p=0..floor(2*(n-3)/3)))/b(n-2), n>=3, 1<= m<= n-2, else 0; with fallfac(q, 3)=A008279(q, 3)=q*(q-1)*(q-2) and b(n)=N(n)/D(n) where D(n) := A090220(n) and N(n) is given in A090220 for n=1..26.

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Original entry on oeis.org

1, 30, 756, 18360, 441936, 10614240, 254788416, 6115201920, 146766525696, 3522406694400, 84537821131776, 2028908069959680, 48693795855814656, 1168651113600245760, 28047626804770062336, 673143043784666480640

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A000400 (powers of 6) with A009968 (powers of 24).

Vincenzo Librandi, Table of n, a(n) for n = 0..730
Index entries for linear recurrences with constant coefficients, signature (30,-144).

Cf. A089505, A089513, A089514.

[6^n*(4^(n+1)-1)/3: n in [0..15]]; // Vincenzo Librandi, Oct 18 2017

CoefficientList[Series[1/((1-6x)(1-24x)),{x,0,20}],x] (* or *) LinearRecurrence[{30,-144},{1,30},20] (* Harvey P. Dale, Sep 25 2017 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)).
a(n) = A089504(n+2, 2), n>=0.
a(n) = (4*(4*3*2)^n - (3*2*1)^n)/3 = (2^n)*(2^(2*(n+1))-1)*3^(n-1).
a(n) = 6^n*(4^(n+1)-1)/3. - Vincenzo Librandi, Oct 18 2017

A089514 Fourth column of triangle A089504.

Original entry on oeis.org

1, 210, 31356, 4150440, 521757936, 64043874720, 7771495098816, 937759335004800, 112842062355914496, 13559707534436743680, 1628284591773850622976, 195461334300256627599360

Offset: 0

Wolfdieter Lang, Dec 01 2003

Convolution of A089513 with powers of 120.

Harvey P. Dale, Table of n, a(n) for n = 0..480
Index entries for linear recurrences with constant coefficients, signature (210,-12744,241920,-1036800).

Cf. A089505, A089513.

CoefficientList[Series[1/((1-6x)(1-24x)(1-60x)(1-120x)),{x,0,20}],x] (* or *) LinearRecurrence[ {210,-12744,241920,-1036800},{1,210,31356,4150440},20] (* Harvey P. Dale, Mar 17 2023 *)

G.f.: 1/((1-3*2*1*x)*(1-4*3*2*x)*(1-5*4*3*x)*(1-6*5*4*x)).
a(n) = A089504(n+4, 4), n>=0.
a(n) = (1350*(6*5*4)^n - 950*(5*4*3)^n + 114*(4*3*2)^n - (3*2*1)^n)/513.

cp's OEIS Frontend

A089506 Denominators used in A089505 to compute column sequences of triangle A089504.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

Extensions

A089504 A generalization of triangle A071951 (Legendre-Stirling).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A089513 Third column of triangle A089504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A089507 Second column of triangle A089504 and second column of array A078741 divided by 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A089514 Fourth column of triangle A089504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula