A145142 -id:A145142 - cp's OEIS frontend

A145149 7th column of A145142.

1, 28, 546, 9450, 165693, 3065238, 59919431, 1226978753, 26377959608, 598190993400, 14328713682920, 361513209493800, 9581318478006976, 266382420824204560, 7761376103890530800, 236610865058490439440, 7532969497593532001856, 250026557590986469841856

Offset: 8

Alois P. Heinz, Oct 03 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 8..100

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 7) *(n-1)!): seq (a(n), n=8..26);

row[n_] := row[n] = Module[{f, i, x, a}, f = Function[Sum[a[i]*#^i, {i, 0, n-1}]]; Function[x, Sum[a[i]*x^i, {i, 0, n-1}] /. First @ Solve[Table[f[i+1] == SeriesCoefficient[x/(1-x-x^4)/(1-x)^i, {x, 0, n}], {i, 0, n-1}]]]]; a[n_] := If[n == 0, 0, SeriesCoefficient[row[n][x], {x, 0, 7}]*(n-1)!]; Table[a[n], {n, 8, 26}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

More terms from Vincenzo Librandi, Feb 15 2014

A145143 1st column of A145142.

Original entry on oeis.org

1, 1, 2, 6, 144, 1200, 9960, 89040, 1231776, 18325440, 280100160, 4415368320, 78497147520, 1538731434240, 32250825734400, 708789321676800, 16531867860480000, 410557135229337600, 10800330695046144000, 298418233851795456000, 8641298765266642944000

Offset: 2

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145142, A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 1) *(n-1)!): seq (a(n), n=2..23);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[2]]*(n-1)!; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145144 2nd column of A145142.

Original entry on oeis.org

1, 3, 11, 50, 634, 6804, 71868, 789984, 11025936, 174509280, 2940903360, 51707242080, 987781034304, 20520063789120, 456583392034560, 10712403843563520, 265316096850923520, 6948996535924162560

Offset: 3

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 2) *(n-1)!): seq (a(n), n=3..23);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[3]]*(n-1)!; Table[a[n], {n, 3, 23}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145145 3rd column of A145142.

Original entry on oeis.org

1, 6, 35, 225, 2464, 29932, 375164, 4877100, 73016856, 1229669496, 22393143552, 430226343456, 8838633396384, 195021406776960, 4592633620285440, 114230969866103040, 2991995263667137536, 82505359191832358400

Offset: 4

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 3) *(n-1)!): seq (a(n), n=4..25);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[4]]*(n-1)!; Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145146 4th column of A145142.

Original entry on oeis.org

1, 10, 85, 735, 8449, 112644, 1605680, 23932700, 391910596, 7073468688, 138120962616, 2862132655200, 62993944853904, 1476042415885824, 36728281476425088, 964322664638298624, 26615080195964032896

Offset: 5

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 4) *(n-1)!): seq (a(n), n=5..25);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[5]]*(n-1)!; Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145147 5th column of A145142.

Original entry on oeis.org

1, 15, 175, 1960, 25473, 375165, 5967170, 99883410, 1794863356, 34897858996, 729308423480, 16183109779200, 379985643499344, 9441352087296912, 247902890514328224, 6852200357138738400, 198716076620554542336

Offset: 6

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 5) *(n-1)!): seq (a(n), n=6..25);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[6]]*(n-1)!; Table[a[n], {n, 6, 25}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145148 6th column of A145142.

Original entry on oeis.org

1, 21, 322, 4536, 68313, 1123815, 19826015, 368232150, 7247538298, 152150838840, 3403471995560, 80589585571040, 2012376195058384, 52929114594971184, 1464737200231998960, 42545324327111272800, 1293727732305595341216

Offset: 7

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 6) *(n-1)!): seq (a(n), n=7..26);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[7]]*(n-1)!; Table[a[n], {n, 7, 26}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145150 8th column of A145142.

Original entry on oeis.org

1, 36, 870, 18150, 369303, 7698834, 166748153, 3751722975, 87886591793, 2152001539688, 55209634265136, 1483339949950248, 41681455251697936, 1223731327819009800, 37510006764224474480, 1199164490827755488960

Offset: 9

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 8) *(n-1)!): seq (a(n), n=9..26);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[9]]*(n-1)!; Table[a[n], {n, 9, 26}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

A145151 9th column of A145142.

Original entry on oeis.org

1, 45, 1320, 32670, 766623, 17990973, 431474615, 10643661600, 271155254513, 7162999744329, 196798229724018, 5629113506142750, 167609902621721416, 5193256923854366136, 167378142642521719832, 5608242214782541676496

Offset: 10

Alois P. Heinz, Oct 03 2008

nonn

Cf. A145153.

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: a:= n-> `if` (n=0, 0, coeftayl (row(n)(x), x=0, 9) *(n-1)!): seq (a(n), n=10..26);

row[n_] := row[n] = Module[{f, a, 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}]; List @@ f[1] /. Solve[eq] // First]; a[n_] := row[n][[10]]*(n-1)!; Table[a[n], {n, 10, 26}] (* Jean-François Alcover, Feb 14 2014, after Maple *)

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)).

Original entry on oeis.org

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)).

cp's OEIS Frontend

A145149 7th column of A145142.

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A145143 1st column of A145142.

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A145144 2nd column of A145142.

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A145145 3rd column of A145142.

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A145146 4th column of A145142.

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A145147 5th column of A145142.

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A145148 6th column of A145142.

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A145150 8th column of A145142.

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A145151 9th column of A145142.

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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)).

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