A049061 -id:A049061 - cp's OEIS frontend

A080856 a(n) = 8n^2 - 4n + 1.

1, 5, 25, 61, 113, 181, 265, 365, 481, 613, 761, 925, 1105, 1301, 1513, 1741, 1985, 2245, 2521, 2813, 3121, 3445, 3785, 4141, 4513, 4901, 5305, 5725, 6161, 6613, 7081, 7565, 8065, 8581, 9113, 9661, 10225, 10805, 11401, 12013, 12641, 13285, 13945, 14621

Offset: 0

Paul Barry, Feb 23 2003

The old definition of this sequence was "Generalized polygonal numbers".
Row T(4,n) of A080853.
{a(k): 0 <= k < 3} = divisors of 25. - Reinhard Zumkeller, Jun 17 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)= coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 27 2010
Also sequence found by reading the segment (1, 5) together with the line from 5, in the direction 5, 25,..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Nov 05 2012
For n > 0: A049061(a(n)) = 0, when the triangle of "signed Eulerian numbers" in A049061 is seen as flattened sequence. - Reinhard Zumkeller, Jan 31 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Reinhard Zumkeller, Enumerations of Divisors.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005408, A000124, A016813, A049061, A080853, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A006261.
A060820 is another version (but the present sequence is the main entry).
A row of the array in A386478.

A080856:=n->8*n^2 - 4*n + 1: seq(A080856(n), n=0..100); # Wesley Ivan Hurt, Jul 16 2017

LinearRecurrence[{3, -3, 1}, {1, 5, 25}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

a(n)=8*n^2-4*n+1 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1+2*x+13*x^2)/(1-x)^3.
a(n) = A060820(n), n>0. - R. J. Mathar, Sep 18 2008
a(n) = C(n,0) + 4*C(n,1) + 16*C(n,2). - Reinhard Zumkeller, Jun 17 2009
a(n) = 16*n+a(n-1)-12 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
E.g.f.: (8*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jun 16 2017

Definition replaced with the closed form by Bruno Berselli, Jan 16 2013

A125300 Tanimoto triangle read by rows: T(n,k) = number of "parity-alternating permutations" (PAPS) of n symbols with k ascents.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 2, 6, 2, 1, 1, 9, 26, 26, 9, 1, 1, 8, 39, 48, 39, 8, 1, 1, 23, 165, 387, 387, 165, 23, 1, 1, 22, 228, 674, 1030, 674, 228, 22, 1, 1, 53, 860, 4292, 9194, 9194, 4292, 860, 53, 1, 1, 52, 1149, 7136, 20738, 28248, 20738, 7136, 1149, 52, 1

Offset: 1

Jonathan Vos Post, Dec 08 2006, Dec 11 2006

A permutation is a parity-alternating permutation (PAP) if its entries take even and odd integers alternately (examples: 436125, 563412 or 7216345). When n is an odd integer, odd entries must appear at both ends of PAPs. T(n,k) = the number of PAPs of {1,2,...,n} with exactly k ascents. Row sums = 2*((n/2)!)^2 if n is even and ((n+1)/2)!*((n-1)/2)! if n is odd.

Arises in combinatorial analysis of signed Eulerian numbers and parity-alternate permutations. This table is the first of three tables on p. 4 of the Tanimoto reference.

			Triangle begins:
n=1.|.1
n=2.|.1....1
n=3.|.1....0....1
n=4.|.1....3....3....1
n=5.|.1....2....6....2....1
n=6.|.1....9...26...26....9....1
n=7.|.1....8...39...48...39....8....1
n=8.|.1...23..165..387..387..165...23....1
n=9.|.1...22..228..674.1030..674..228...22....1
n=10|.1...53..860.4292.9194.9194.4292..860...53....1
Examples of parity-alternating permutations of n=5 and their number of rises k are [1,2,3,4,5] (k=4, only rises), [1,2,5,4,3] (k=2: 1->2 and 2->5), [1,4,3,2,5] (k=2: 1->4 and 2->5).  The T(n=5,k=1)=2 parity-alternating permutations with k=1 rise are [3,2,5,4,1] and [5,2,1,4,3].

Alois P. Heinz, Rows n = 1..27, flattened
Shinji Tanimoto, Alternate Permutations and Signed Eulerian Numbers, math.CO/0612135, Ann. Comb. 14 (2010), 355.

Cf. A008292 = Triangle of Eulerian numbers T(n, k) read by rows, A049061 = Triangle a(n, k) (1<=k<=n) of signed Eulerian numbers.

Row sums give: A092186. - Alois P. Heinz, Nov 18 2013

isPAP := proc(per) local i ; for i from 2 to nops(per) do if ( op(i,per) mod 2 ) = (op(i-1,per) mod 2 ) then RETURN(false) ; fi ; od : RETURN(true) ; end: ascents := proc(per) local i, asc ; asc :=0 ; for i from 2 to nops(per) do if op(i,per) > op (i-1,per) then asc := asc+1 : fi ; od : RETURN(asc) ; end:
A125300row := proc(n) local per,resul, asc,thisp,p,i,row ; row := array(0..n-1) ; for i from 0 to n-1 do row[i] := 0 : od ; per := combinat[permute](n) ; for p from 1 to nops(per) do asc := 0 ; thisp := op(p,per) ; if isPAP(thisp) then asc := ascents(thisp) ; row[asc] := row[asc]+1 ; fi ; od ; RETURN(row) : end: for n from 2 to 10 do r := A125300row(n) ; for k from 0 to n-1 do print(r[k]) ; od : od : # R. J. Mathar, Dec 12 2006

isPAP[per_] := (For[i = 2, i <= Length[per], i++, If [Mod[per[[i]], 2] == Mod[per[[i - 1]], 2], Return[False] ] ]; True);
ascents[per_] := (asc = 0; For[i = 2, i <= Length[per], i++, If[per[[i]] > per[[i - 1]],  asc ++] ]; asc);
A125300row[n_] := (row = Range[0, n - 1]; For[i = 0, i <= n - 1, i++, row[[i]] = 0]; per = Permutations[Range[n]]; For[p = 1, p <= Length[per], p++, asc = 0; thisp = per[[p]]; If[isPAP[thisp], asc = ascents[thisp]; row[[asc]] += 1]]; row);
Join[{1}, Reap[For[n = 2, n <= 10, n++, r = A125300row[n]; For[k = 0, k <= n - 1, k++, Print[r[[k]]]; Sow[r[[k]]]]]][[2, 1]]] (* Jean-François Alcover, Nov 07 2017, after R. J. Mathar's Maple code *)

T(n,k) = T(n,n-k-1).

Corrected by R. J. Mathar, Dec 12 2006
Edited by N. J. A. Sloane, Dec 21 2006
Replaced arXiv URL by non-cached version - R. J. Mathar, Oct 30 2009
More terms from Alois P. Heinz, Nov 18 2013

A128612 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 5, 5, 1, 1, 14, 30, 14, 1, 0, 28, 155, 147, 29, 1, 0, 56, 605, 1208, 586, 64, 1, 1, 127, 2133, 7819, 7819, 2133, 127, 1, 1, 262, 7288, 44074, 78190, 44074, 7288, 262, 1, 0, 496, 23947, 227623, 655039, 655315, 227569, 23893, 517, 1, 0, 992, 76305, 1102068, 4868556, 7862124, 4869558, 1101420, 76332, 1044, 1

Offset: 1

Ralf Stephan, May 08 2007

			Triangle starts:
  1;
  0,   1;
  0,   2,    1;
  1,   5,    5,    1;
  1,  14,   30,   14,    1;
  0,  28,  155,  147,   29,    1;
  0,  56,  605, 1208,  586,   64,   1;
  1, 127, 2133, 7819, 7819, 2133, 127, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, On the joint distribution of descents and signs of permutations, arXiv:1910.04258 [math.CO], 2019.
S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006.

Cf. A008292, A049061, A128613.
Cf. A145882 (similar with rows reversed).
Row sums give A001710.
T(2n,n) gives A382309.

A008292 := proc(n,k) local j; add( (-1)^j*(k-j)^n*binomial(n+1,j),j=0..k) ; end: A049061 := proc(n,k) if k <= 0 or n <=0 or k > n then 0; elif n = 1 then 1 ; elif n mod 2 = 0 then A049061(n-1,k)-A049061(n-1,k-1) ; else k*A049061(n-1,k)+(n-k+1)*A049061(n-1,k-1) ; fi ; end: A128612 := proc(n,k) (A008292(n,n-k)+A049061(n,n-k))/2 ; end: for n from 1 to 11 do for k from 0 to n-1 do printf("%d,",A128612(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007
# second Maple program:
b:= proc(u, o, i) option remember; expand(`if`(u+o=0, 1-i,
       add(b(u+j-1, o-j, irem(i+u+j-1, 2)), j=1..o)*x+
       add(b(u-j, o+j-1, irem(i+u-j, 2)), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, May 02 2017

b[u_, o_, i_] := b[u, o, i] = Expand[If[u + o == 0, 1 - i, Sum[b[u + j - 1, o - j, Mod[i + u + j - 1, 2]], {j, 1, o}]*x + Sum[b[u - j, o + j - 1, Mod[i + u - j, 2]], {j, 1, u}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0,0]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

T(n,k) = (1/2) * (A008292(n,n-k) + A049061(n,n-k)), n>=1, 0<=kR. J. Mathar, Nov 01 2007

More terms from R. J. Mathar, Nov 01 2007

A373657 Triangle read by rows: Coefficients of the polynomials P(n, x) * EP(n, x), where P denote the signed Pascal polynomials and EP the Eulerian polynomials A173018.

Original entry on oeis.org

1, -1, 1, 1, -1, -1, 1, -1, -1, 8, -8, 1, 1, 1, 7, -27, 19, 19, -27, 7, 1, -1, -21, 54, 54, -276, 276, -54, -54, 21, 1, 1, 51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1, -1, -113, -372, 3436, -5125, -5013, 21216, -21216, 5013, 5125, -3436, 372, 113, 1

Offset: 0

Peter Luschny, Jun 15 2024

			Triangle starts:
[0] [ 1]
[1] [-1,   1]
[2] [ 1,  -1,  -1,    1]
[3] [-1,  -1,   8,   -8,    1,     1]
[4] [ 1,   7, -27,   19,   19,   -27,     7,    1]
[5] [-1, -21,  54,   54, -276,   276,   -54,  -54,   21,   1]
[6] [ 1,  51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1]

S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006. (see p. 7)

Cf. A173018, A049061, A101842, A000007 (row sums).

PolyProd := proc(P, Q, len) local ep, eq, epq, CL, n, k;
ep := (n, x) -> simplify(add(Q(n, k)*x^k, k = 0..n)):
eq := (n, x) -> simplify(add(P(n, k)*x^k, k = 0..n)):
epq := (n, x) -> expand(ep(n, x) * eq(n, x)):
CL := p -> PolynomialTools:-CoefficientList(p, x);
seq(CL(epq(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
PolyProd((n, k) -> (-1)^(n-k)*binomial(n, k), combinat:-eulerian1, 7);

A128613 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an odd number of inversions.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 0, 6, 6, 0, 0, 12, 36, 12, 0, 1, 29, 147, 155, 28, 0, 1, 64, 586, 1208, 605, 56, 0, 0, 120, 2160, 7800, 7800, 2160, 120, 0, 0, 240, 7320, 44160, 78000, 44160, 7320, 240, 0, 1, 517, 23893, 227569, 655315, 655039, 227623, 23947, 496, 0, 1, 1044, 76332, 1101420, 4869558, 7862124, 4868556, 1102068, 76305, 992, 0

Offset: 1

Ralf Stephan, May 08 2007

			Triangle starts:
  0;
  1,   0;
  1,   2,    0;
  0,   6,    6,    0;
  0,  12,   36,   12,    0;
  1,  29,  147,  155,   28,    0;
  1,  64,  586,  120,  605,   56,   0;
  0, 120, 2160, 7800, 7800, 2160, 120, 0;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Jason Fulman, Gene B. Kim, Sangchul Lee, and T. Kyle Petersen, On the joint distribution of descents and signs of permutations, arXiv:1910.04258 [math.CO], 2019.
S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006.

Cf. A008292, A049061, A128612.

A008292 := proc(n,k) local j; add( (-1)^j*(k-j)^n*binomial(n+1,j),j=0..k) ; end: A049061 := proc(n,k) if k <= 0 or n <=0 or k > n then 0; elif n = 1 then 1 ; elif n mod 2 = 0 then A049061(n-1,k)-A049061(n-1,k-1) ; else k*A049061(n-1,k)+(n-k+1)*A049061(n-1,k-1) ; fi ; end: A128613 := proc(n,k) (A008292(n,n-k)-A049061(n,n-k))/2 ; end: for n from 1 to 11 do for k from 0 to n-1 do printf("%d,",A128613(n,k)) ; od: od: # R. J. Mathar, Nov 01 2007
# second Maple program:
b:= proc(u, o, i) option remember; expand(`if`(u+o=0, i,
       add(b(u+j-1, o-j, irem(i+u+j-1, 2)), j=1..o)*x+
       add(b(u-j, o+j-1, irem(i+u-j, 2)), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, May 02 2017

b[u_, o_, i_] := b[u, o, i] = Expand[If[u + o == 0, i, Sum[b[u + j - 1, o - j, Mod[i + u + j - 1, 2]], {j, 1, o}]*x + Sum[b[u - j, o + j - 1, Mod[i + u - j, 2]], {j, 1, u}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0,0]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

a(n) = 1/2 * (A008292(n,k) - A049061(n,k)).

T(n,k) = 1/2 * (A008292(n,n-k) - A049061(n,n-k)), n>=1, 0<=kR. J. Mathar, Nov 01 2007

Corrected and extended by R. J. Mathar, Nov 01 2007

A263985 Triangle of signed Eulerian numbers on involutions, read by rows.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 1, -2, -2, 1, 1, 6, 0, -2, 1, -1, 3, 14, 2, -3, 1, -1, -12, -15, 12, -1, -4, 1, 1, -4, -51, -76, 4, -3, -4, 1, 1, 20, 67, -10, -80, 30, 3, -4, 1, -1, 5, 137, 517, 414, 66, 75, 7, -5, 1, -1, -30, -192, -140, 721, 588, -49, 44, 0, -6, 1

Offset: 1

Michel Marcus, Oct 31 2015

			Triangle begins:
1;
-1, 1;
-1, -2, 1;
1, -2, -2, 1;
1, 6, 0, -2, 1;
-1, 3, 14, 2, -3, 1;
-1, -12, -15, 12, -1, -4, 1;
...

M. Barnabei, F. Bonetti, M. Silimbani, The signed Eulerian numbers on involutions, PU. M. A. Vol. 19 (2008) pp. 117-126.
M. Barnabei, F. Bonetti, M. Silimbani, The signed Eulerian numbers on involutions, arXiv:0803.2126 [math.CO], 2008.
J. Desarmenien and D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992), no. 1-3, 49-58.

Cf. A049061.

T[n_, k_] := Sum[(-1)^(k-m+1) Binomial[n+1, k-m+1] Sum[(-1)^j Binomial[ Binomial[m+1, 2]+j-1, j] Binomial[m, n-2j], {j, 0, n/2}], {m, 0, k+1}];
Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 26 2018 *)

T(n, k) = sum(m=0, k+1, (-1)^(k-m+1)*binomial(n+1,k-m+1)*sum(j=0,n\2, (-1)^j*binomial(binomial(m+1,2)+j-1,j)*binomial(m,n-2*j)));

T(n, k) = Sum_{m=0..k+1} (-1)^(k-m+1)*C(n+1,k-m+1)*Sum_{j=0..floor(n/2)} (-1)^j*C(C(m+1,2)+j-1,j)*C(m,n-2*j);

cp's OEIS Frontend

A080856 a(n) = 8*n^2 - 4*n + 1.

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A125300 Tanimoto triangle read by rows: T(n,k) = number of "parity-alternating permutations" (PAPS) of n symbols with k ascents.

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A128612 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an even number of inversions.

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A373657 Triangle read by rows: Coefficients of the polynomials P(n, x) * EP(n, x), where P denote the signed Pascal polynomials and EP the Eulerian polynomials A173018.

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Maple

A128613 Triangle T(n,k) read by rows: number of permutations in [n] with exactly k ascents that have an odd number of inversions.

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Examples

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Programs

Maple

Mathematica

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Extensions

A263985 Triangle of signed Eulerian numbers on involutions, read by rows.

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Author

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Examples

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Programs

Mathematica

PARI

Formula

A080856 a(n) = 8n^2 - 4n + 1.