A145141 -id:A145141 - 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.

A145126 a(n) = 1 + (6 + (11 + (6 + n)n)n)*n/24.

Original entry on oeis.org

1, 2, 6, 16, 36, 71, 127, 211, 331, 496, 716, 1002, 1366, 1821, 2381, 3061, 3877, 4846, 5986, 7316, 8856, 10627, 12651, 14951, 17551, 20476, 23752, 27406, 31466, 35961, 40921, 46377, 52361, 58906, 66046, 73816, 82252, 91391, 101271, 111931, 123411, 135752

Offset: 0

Alois P. Heinz, Oct 03 2008

From Gary W. Adamson, Jul 31 2010: (Start)
Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 6, 10, 15, ...).
Example: a(4) = 36 = (5, 4, 3, 2, 1) dot (1, 0, 3, 6, 10) = (5 + 0 + 9 + 12 + 10). (End)
Also the number of permutations of length n that can be sorted by a single block interchange (in the sense of Christie). - Vincent Vatter, Aug 21 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. A. Christie, Sorting Permutations by Block-Interchanges, Inf. Process. Lett. 60 (1996), 165-169.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
C. Homberger and V. Vatter, On the effective and automatic enumeration of polynomial permutation classes. [Broken link]
C. Homberger, V. Vatter, On the effective and automatic enumeration of polynomial permutation classes, arXiv preprint arXiv:1308.4946 [math.CO], 2013-2015.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

5th row of A145153. See row 5 of A145140/A145141 for rational coefficients and A145142 for 24 * coefficients of polynomial.

a:= n-> 1+ (6+ (11+ (6+ n) *n) *n) *n/24: seq(a(n), n=0..40);
# second Maple program:
with(combinat): seq(binomial(n+3, 4)+1, n=0..40); # Zerinvary Lajos, Mar 24 2009

a=b=s=0;lst={a};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,6!}];lst+1 (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
CoefficientList[Series[(x^4 - 4 x^3 + 6 x^2 - 3 x + 1) / (1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

Vec((x^4-4*x^3+6*x^2-3*x+1)/(1-x)^5 + O(x^50)) \\ Altug Alkan, Nov 24 2015

G.f.: (x^4-4*x^3+6*x^2-3*x+1) / (1-x)^5.

a(n) = C(n+3,4)+1. - Zerinvary Lajos, Mar 24 2009

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.

A145127 a(n) = 1 + (144 + (50 + (35 + (10 + n)n)n)n)n/120.

Original entry on oeis.org

1, 3, 9, 25, 61, 132, 259, 470, 801, 1297, 2013, 3015, 4381, 6202, 8583, 11644, 15521, 20367, 26353, 33669, 42525, 53152, 65803, 80754, 98305, 118781, 142533, 169939, 201405, 237366, 278287, 324664, 377025, 435931, 501977, 575793, 658045, 749436

Offset: 0

Alois P. Heinz, Oct 03 2008

nonn

(1 + 3x + 9x^2 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x^2 + 6x^3 + 10x^4 + ...). - Gary W. Adamson, Jul 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

6th row of A145153. See row 6 of A145140/A145141 for rational coefficients and A145142 for 120 * coefficients of polynomial.

[1 + (144 + (50 + (35 + (10 + n)*n)*n)*n)*n/120: n in [0..40]]; // Vincenzo Librandi, May 19 2011

a := n-> 1+ (144+ (50+ (35+ (10+ n) *n) *n) *n) *n/120: seq (a(n), n=0..40);

G.f.: (x^4 - 4*x^3 + 6*x^2 - 3*x + 1) / (1-x)^6.

A145128 a(n) = 1 + (1200 + (634 + (225 + (85 + (15 + n)n)n)n)n)*n/720.

Original entry on oeis.org

1, 4, 13, 38, 99, 231, 490, 960, 1761, 3058, 5071, 8086, 12467, 18669, 27252, 38896, 54417, 74784, 101137, 134806, 177331, 230483, 296286, 377040, 475345, 594126, 736659, 906598, 1108003, 1345369, 1623656, 1948320, 2325345, 2761276, 3263253, 3839046, 4497091

Offset: 0

Alois P. Heinz, Oct 03 2008

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

7th row of A145153. See row 7 of A145140/A145141 for rational coefficients and A145142 for 720 * coefficients of polynomial.

a := n-> 1+ (1200+ (634+ (225+ (85+ (15+ n) *n) *n) *n) *n) *n/720: seq (a(n), n=0..40);

G.f.: (x^4-4*x^3+6*x^2-3*x+1) / (1-x)^7.

A145129 1 + (9960 + (6804 + (2464 + (735 + (175 + (21 + n)n)n)n)n)n)n/5040.

Original entry on oeis.org

1, 5, 18, 56, 155, 386, 876, 1836, 3597, 6655, 11726, 19812, 32279, 50948, 78200, 117096, 171513, 246297, 347434, 482240, 659571, 890054, 1186340, 1563380, 2038725, 2632851, 3369510, 4276108, 5384111, 6729480, 8353136, 10301456, 12626801, 15388077, 18651330

Offset: 0

Alois P. Heinz, Oct 03 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

8th row of A145153. See row 8 of A145140/A145141 for rational coefficients and A145142 for 5040 * coefficients of polynomial.

a := n-> 1+ (9960+ (6804+ (2464+ (735+ (175+ (21+ n) *n) *n) *n) *n) *n) *n/5040: seq(a(n), n=0..40);

CoefficientList[Series[(x^4 - 4 x^3 + 6 x^2 - 3 x + 1) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

G.f.: (x^4-4*x^3+6*x^2-3*x+1) / (1-x)^8.

A145130 2 + (89040 + (71868 + (29932 + (8449 + (1960 + (322 + (28 + n)n)n)n)n)n)n)*n/40320.

Original entry on oeis.org

2, 7, 25, 81, 236, 622, 1498, 3334, 6931, 13586, 25312, 45124, 77403, 128351, 206551, 323647, 495160, 741457, 1088891, 1571131, 2230702, 3120756, 4307096, 5870476, 7909201, 10542052, 13911562, 18187670, 23571781, 30301261, 38654397, 48955853, 61582654

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 (9, -36, 84, -126, 126, -84, 36, -9, 1).

9th row of A145153. See row 9 of A145140/A145141 for rational coefficients and A145142 for 40320 * coefficients of polynomial.

a:= n-> 2+ (89040+ (71868+ (29932+ (8449+ (1960+ (322+ (28+ n) *n) *n) *n) *n) *n) *n) *n/40320: seq (a(n), n=0..40);

Table[2+(89040+(71868+(29932+(8449+(1960+(322+(28+n)n)n)n)n)n)n)n/40320,{n,0,40}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{2,7,25,81,236,622,1498,3334,6931},40](* Harvey P. Dale, Dec 25 2011 *)
CoefficientList[Series[(x^8 - 8 x^7 + 28 x^6 - 56 x^5 + 71 x^4 - 60 x^3 + 34 x^2 - 11 x + 2) / (1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 06 2013 *)

a(n)=2+(89040+(71868+(29932+(8449+(1960+(322+(28+n)*n)*n)*n)*n)*n)*n)*n/40320 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (x^8-8*x^7+28*x^6-56*x^5+71*x^4-60*x^3+34*x^2-11*x+2) / (1-x)^9.

a(0)=2, a(1)=7, a(2)=25, a(3)=81, a(4)=236, a(5)=622, a(6)=1498, a(7)=3334, a(8)=6931, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3) -126*a(n-4) +126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). - Harvey P. Dale, Dec 25 2011

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)).

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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.

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A145126 a(n) = 1 + (6 + (11 + (6 + n)*n)*n)*n/24.

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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.

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A145127 a(n) = 1 + (144 + (50 + (35 + (10 + n)*n)*n)*n)*n/120.

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A145128 a(n) = 1 + (1200 + (634 + (225 + (85 + (15 + n)*n)*n)*n)*n)*n/720.

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A145129 1 + (9960 + (6804 + (2464 + (735 + (175 + (21 + n)*n)*n)*n)*n)*n)*n/5040.

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A145130 2 + (89040 + (71868 + (29932 + (8449 + (1960 + (322 + (28 + n)*n)*n)*n)*n)*n)*n)*n/40320.

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A145126 a(n) = 1 + (6 + (11 + (6 + n)n)n)*n/24.

A145127 a(n) = 1 + (144 + (50 + (35 + (10 + n)n)n)n)n/120.

A145128 a(n) = 1 + (1200 + (634 + (225 + (85 + (15 + n)n)n)n)n)*n/720.

A145129 1 + (9960 + (6804 + (2464 + (735 + (175 + (21 + n)n)n)n)n)n)n/5040.

A145130 2 + (89040 + (71868 + (29932 + (8449 + (1960 + (322 + (28 + n)n)n)n)n)n)n)*n/40320.