A145130 -id:A145130 - cp's OEIS frontend

A145153 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)).

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

Offset: 0

Alois P. Heinz, Oct 03 2008

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.

			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, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows 0-9 give: A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.
Columns 0-9 give: A017898(n-1) for n>0, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.
Main diagonal gives: A145138.
Antidiaginal sums give: A145139.
Numerators/denominators of polynomials for rows give: A145140/A145141.
Cf. A145142, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150, A145151, A145152.

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);

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 *)

G.f. of column k: x/((1-x-x^4)*(1-x)^(k-1)).

A145142 Triangle T(n,k), n>=1, 0<=k<=n-1, read by rows: T(n,k)/(n-1)! is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 24, 6, 11, 6, 1, 120, 144, 50, 35, 10, 1, 720, 1200, 634, 225, 85, 15, 1, 5040, 9960, 6804, 2464, 735, 175, 21, 1, 80640, 89040, 71868, 29932, 8449, 1960, 322, 28, 1, 1088640, 1231776, 789984, 375164, 112644, 25473, 4536, 546, 36, 1

Offset: 1

Alois P. Heinz, Oct 03 2008

			Triangle begins:
    1;
    0,   1;
    0,   1,   1;
    0,   2,   3,   1;
   24,   6,  11,   6,   1;
  120, 144,  50,  35,  10,  1;

Alois P. Heinz, Rows n = 1..45, flattened

T(n,k)/(n-1)! gives: A145140 / A145141.
Columns 0-9 give: A052581, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150.
Diagonal and lower diagonals 1-3 give: A000012, A000217, A000914, A001303.
Cf. A145153, A001477, A000292, A145126, A145127, A145128, A145129, A145130.
Row sums are in A052593.

row:= proc(n) option remember; local f,i,x; f:= unapply(simplify(sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); unapply(subs(solve({seq(f(i+1)= coeftayl(x/ (1-x-x^4)/ (1-x)^i, x=0, n), i=0..n-1)}, {seq(cat(a||i), i=0..n-1)}), sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); end: T:= (n,k)-> `if`(k<0 or k>=n,0, coeff(row(n)(x),x,k)*(n-1)!): seq(seq(T(n,k), k=0..n-1), n=1..12);

row[n_] := Module[{f, eq}, f = Function[x, Sum[a[k]*x^k, {k, 0, n-1}]]; eq = Table[f[k+1] == SeriesCoefficient[x/(1-x-x^4)/(1-x)^k, {x, 0, n}], {k, 0, n-1}]; Table[a[k], {k, 0, n-1}] /. Solve[eq] // First]; Table[row[n]*(n-1)!, {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 04 2014, after Alois P. Heinz *)

See program.

A145140 Numerators of triangle T(n,k), n>=1, 0<=k<=n - 1, read by rows: T(n,k) is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 11, 1, 1, 1, 6, 5, 7, 1, 1, 1, 5, 317, 5, 17, 1, 1, 1, 83, 27, 22, 7, 5, 1, 1, 2, 53, 5989, 1069, 1207, 7, 23, 1, 1, 3, 611, 2743, 93791, 149, 1213, 1, 13, 1, 1, 4, 101, 25523, 5419, 20071, 397, 3253, 1, 29, 1, 1, 5, 32419, 11017, 30731, 21757

Offset: 1

Alois P. Heinz, Oct 03 2008

			1, 0, 1, 0, 1/2, 1/2, 0, 1/3, 1/2, 1/6, 1, 1/4, 11/24, 1/4, 1/24, 1, 6/5, 5/12, 7/24, 1/12, 1/120, 1, 5/3, 317/360, 5/16, 17/144, 1/48, 1/720 ... = A145140/A145141
As triangle:
  1
  0 1
  0 1/2 1/2
  0 1/3 1/2 1/6
  1 1/4 11/24 1/4 1/24
  1 6/5 5/12 7/24 1/12 1/120

Denominators of T(n, k): A145141. T(n, k)*(n-1)!: A145142.
Row sums give: A003269, A017898(n+3).
Cf. A145153, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.

row:= proc(n) option remember; local f,i,x; f:= unapply(simplify(sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); unapply(subs(solve({seq(f(i+1)= coeftayl(x/ (1-x-x^4)/ (1-x)^i, x=0, n), i=0..n-1)}, {seq(cat(a||i), i=0..n-1)}), sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); end: T:= (n,k)-> coeff(row(n)(x), x, k): seq(seq(numer(T(n,k)), k=0..n-1), n=1..14);

row[n_] := Module[{f, eq}, f = Function[x, Sum[a[k]*x^k, {k, 0, n-1}]]; eq = Table[f[k+1] == SeriesCoefficient[x/(1-x-x^4)/(1-x)^k, {x, 0, n}], {k, 0, n-1}]; Table[a[k], {k, 0, n-1}] /. Solve[eq] // First]; Table[row[n] // Numerator, {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 04 2014, after Alois P. Heinz *)

See program.

A145138 Main diagonal of square array A145153.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Oct 03 2008

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A145153, A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.

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:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 18 2019

a[n_] := SeriesCoefficient[x/(1-x-x^4)/(1-x)^(n-1), {x, 0, n}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 10 2022 *)

a(n) = [x^n] x/((1-x-x^4)*(1-x)^(n-1)).

A145139 Antidiagonal sums of A145153.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Oct 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,1,-2).

Cf. A145153, A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130, A017898, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.

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

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-x)^2/((1-2*x)*(1-x-x^4)) )); // G. C. Greubel, May 21 2019

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);

CoefficientList[Series[x*(1-x)^2/((1-2*x)*(1-x-x^4)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)

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

(x*(1-x)^2/((1-2*x)*(1-x-x^4))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 21 2019

G.f.: x*(1-x)^2 / ((1-2*x)*(1-x-x^4)).

cp's OEIS Frontend

A145153 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)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A145142 Triangle T(n,k), n>=1, 0<=k<=n-1, read by rows: T(n,k)/(n-1)! is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A145140 Numerators of triangle T(n,k), n>=1, 0<=k<=n - 1, read by rows: T(n,k) is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A145138 Main diagonal of square array A145153.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A145139 Antidiagonal sums of A145153.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula