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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where sequence a_k of column k is the expansion of x/((1 - x - x^4)*(1 - x)^(k - 1)).
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28
0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 1, 0, 1, 4, 6, 4, 2, 1, 0, 1, 5, 10, 10, 6, 3, 1, 0, 1, 6, 15, 20, 16, 9, 4, 1, 0, 1, 7, 21, 35, 36, 25, 13, 5, 2, 0, 1, 8, 28, 56, 71, 61, 38, 18, 7, 3, 0, 1, 9, 36, 84, 127, 132, 99, 56, 25, 10, 4, 0, 1, 10, 45, 120, 211, 259, 231, 155, 81, 35, 14, 5
COMMENTS
Each row sequence a_n (for n > 0) is produced by a polynomial of degree n-1, whose (rational) coefficients are given in row n of A145140/ A145141. The coefficients *(n-1)! are given in A145142. Each column sequence a_k is produced by a recursion, whose coefficients are given by row k of A145152.
FORMULA
G.f. of column k: x/((1-x-x^4)*(1-x)^(k-1)).
EXAMPLE
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 1, 4, 10, 20, 35, 56, ...
1, 2, 6, 16, 36, 71, 127, ...
MAPLE
A:= proc(n, k) coeftayl (x/ (1-x-x^4)/ (1-x)^(k-1), x=0, n) end:
seq(seq(A(n, d-n), n=0..d), d=0..13);
MATHEMATICA
a[n_, k_] := SeriesCoefficient[x/(1 - x - x^4)/(1 - x)^(k - 1), {x, 0, n}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 05 2013 *)
CROSSREFS
Numerators/denominators of polynomials for rows give: A145140/ A145141. Cf. A145142, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150, A145151, A145152.
Triangle T(k,m), k>= - 3, 1<=m<=k + 4, read by rows: T(k,m) is the coefficient of a_k(n - m) in the recursive evaluation of a_k(n), where a_k is the k - th column sequence of A145153.
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0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 2, -1, 0, 1, -1, 0, 3, -3, 1, 1, -2, 1, 0, 4, -6, 4, 0, -3, 3, -1, 0, 5, -10, 10, -4, -3, 6, -4, 1, 0, 6, -15, 20, -14, 1, 9, -10, 5, -1, 0, 7, -21, 35, -34, 15, 8, -19, 15, -6, 1, 0, 8, -28, 56, -69, 49, -7, -27, 34, -21, 7, -1, 0, 9, -36, 84
COMMENTS
Rows -3<=k<0 are used to form the shape of a perfect triangle.
EXAMPLE
Triangle begins:
0;
0, 0;
0, 0, 0;
1, 0, 0, 1;
1, 0, 0, 1, 0;
2, -1, 0, 1, -1, 0;
3, -3, 1, 1, -2, 1, 0;
MAPLE
cor:= proc(k) option remember; `if`(k<=0, [1, 0, 0, 1], [seq(coeff( -(1-x-x^4) *(1-x)^(k-1), x, j), j=1..k+3)]) end: T:= proc(k, m) local l, j; l:= cor(k); `if`(k<0 or m<0 or m>nops(l), 0, l[m]) end: seq(seq(T(k, m), m=1..k+4), k=-3..13);
MATHEMATICA
cor[k_] := cor[k] = If[k <= 0, {1, 0, 0, 1}, Table[Coefficient[-(1-x-x^4)*(1-x)^(k-1), x, j], {j, 1, k+3}]]; T[k_, m_] := Module[{l, j}, l = cor[k]; If[k<0 || m<0 || m>Length[l], 0, l[[m]]]]; Table[Table[T[k, m], {m, 1, k+4}], {k, -3, 13}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)
CROSSREFS
Cf. A145153, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.
Main diagonal of square array A145153.
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2
0, 1, 2, 6, 20, 71, 259, 960, 3597, 13586, 51635, 197223, 756380, 2910707, 11233311, 43460144, 168502849, 654547456, 2546819347, 9924285801, 38723794820, 151278566731, 591628491483, 2316065644414, 9074988880769, 35587925333525, 139666503235814, 548516611541343
FORMULA
a(n) = [x^n] x/((1-x-x^4)*(1-x)^(n-1)).
MAPLE
a:= n-> coeftayl(x/(1-x-x^4)/(1-x)^(n-1), x=0, n):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, n*(n+1)*(n^2-4*n+6)/6,
a(n-4)+(2*(35*n^3-207*n^2+310*n-78)*a(n-1)-(203*n^3
-1244*n^2+1891*n-130)*a(n-2)+(2*n-7)*(7*n-19)*n*
(10*a(n-3)-2*a(n-5)))/((7*n-26)*(n-1)^2))
end:
MATHEMATICA
a[n_] := SeriesCoefficient[x/(1-x-x^4)/(1-x)^(n-1), {x, 0, n}];
CROSSREFS
Cf. A145153, A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.
0, 1, 1, 2, 4, 9, 18, 36, 72, 145, 291, 583, 1167, 2336, 4675, 9354, 18713, 37433, 74876, 149766, 299551, 599128, 1198292, 2396634, 4793337, 9586769, 19173669, 38347519, 76695288, 153390921, 306782318, 613565293, 1227131493, 2454264238
FORMULA
G.f.: x*(1-x)^2 / ((1-2*x)*(1-x-x^4)).
MAPLE
a:= n-> (Matrix([[4, 2, 1, 1, 0]]). Matrix (5, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1, -2][i] else 0 fi)^n)[1, 5]: seq(a(n), n=0..40);
MATHEMATICA
CoefficientList[Series[x*(1-x)^2/((1-2*x)*(1-x-x^4)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
PROG
(PARI) my(x='x+O('x^40)); concat([0], Vec(x*(1-x)^2/((1-2*x)*(1-x-x^4)))) \\ G. C. Greubel, May 21 2019 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-x)^2/((1-2*x)*(1-x-x^4)) )); // G. C. Greubel, May 21 2019 (Sage) (x*(1-x)^2/((1-2*x)*(1-x-x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 21 2019 (GAP) a:=[0, 1, 1, 2, 4];; for n in [6..40] do a[n]:=3*a[n-1]-2*a[n-2]+a[n-4] -2*a[n-5]; od; a; # G. C. Greubel, May 21 2019
CROSSREFS
Cf. A145153, A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.
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