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A136215
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Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.
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3
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1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Comments from Peter Bala (pbala(AT)toucansurf.com), Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalised Pascal triangle P(a,b), a lower unit triangular matrix, shown below
n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216.
T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1). E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)
The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). [From Tom Copeland (tcjpn(AT)msn.com), Aug 18 2008]
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FORMULA
| Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
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EXAMPLE
| Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and
column k of V = column 0 of T^(3k+2) where V = A112333.
This triangle T begins:
1;
1, 1;
4, 2, 1;
28, 12, 3, 1;
280, 112, 24, 4, 1;
3640, 1400, 280, 40, 5, 1;
58240, 21840, 4200, 560, 60, 6, 1;
1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
Triangle U = A136214 begins:
1;
1, 1;
4, 4, 1;
28, 28, 7, 1;
280, 280, 70, 10, 1;
3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0.
Triangle V = A112333 begins:
1;
2, 1;
10, 5, 1;
80, 40, 8, 1;
880, 440, 88, 11, 1;
12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.
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PROG
| (PARI) T(n, k)=binomial(n, k)*if(n-k==0, 1, prod(j=0, n-k-1, 3*j+1))
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CROSSREFS
| Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.
Cf. A094587.
Sequence in context: A158835 A138271 A136212 * A136737 A004551 A016511
Adjacent sequences: A136212 A136213 A136214 * A136216 A136217 A136218
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2008
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