A205497 -id:A205497 - cp's OEIS frontend

A373432 Triangle read by rows. Coefficients of the polynomials P(n, x) * EZ(n, x), where P denote the Pascal polynomials and EZ the zig-zag Eulerian polynomials A205497.

1, 1, 1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 7, 19, 26, 19, 7, 1, 1, 12, 52, 116, 150, 116, 52, 12, 1, 1, 20, 130, 430, 845, 1052, 845, 430, 130, 20, 1, 1, 33, 312, 1453, 4023, 7218, 8736, 7218, 4023, 1453, 312, 33, 1, 1, 54, 730, 4639, 17316, 42142, 70593, 83610, 70593, 42142, 17316, 4639, 730, 54, 1

Offset: 0

Peter Luschny, Jun 05 2024

There are various conventions for indexing Eulerian numbers. The one used here is described by the condition that for all polynomials p(n, 0) = 1. This applies equally to the classical Eulerian polynomials given by the coefficients A173018, as well as to the Eulerian zig-zag polynomials with coefficients in A205497 and to the polynomials here. See the illustration (link section).

			Triangle starts:
  [0] [1]
  [1] [1,  1]
  [2] [1,  2,   1]
  [3] [1,  4,   6,   4,   1]
  [4] [1,  7,  19,  26,  19,    7,   1]
  [5] [1, 12,  52, 116, 150,  116,  52,  12,   1]
  [6] [1, 20, 130, 430, 845, 1052, 845, 430, 130, 20, 1]

Peter Luschny, Illustrating the polynomials.

Cf. A000831 (row sums), A007318 (Pascal), A205497 (zig-zag Eulerian).

EZP := proc(P, len) local R, EZ, EP, EZP, CL, n;
R := proc(n) option remember; local F; if n = 0 then 1/(1-q*x) else F := R(n-1);
simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1-x)^(n+1))/x^2)):
EP := (n, x) -> local k; simplify(add(P(n, k)*x^k, k = 0..n)):
EZP := (n, x) -> expand(EZ(n, x) * EP(n, x)):
CL := p -> PolynomialTools:-CoefficientList(p, x);
seq(CL(EZP(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
EZP(binomial, 8);

A373428 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EZ(n, x), where S2 denote the Stirling set polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 4, 1, 0, 1, 10, 28, 26, 9, 1, 0, 1, 22, 137, 291, 261, 102, 17, 1, 0, 1, 45, 555, 2300, 4150, 3517, 1479, 306, 29, 1, 0, 1, 89, 2048, 15152, 48942, 76259, 61846, 26976, 6388, 795, 47, 1

Offset: 0

Peter Luschny, Jun 06 2024

			Tracing the computation:
0: [1] *          [1] =                   [1]
1: [1] *          [0, 1] =                [0, 1]
2: [1] *          [0, 1, 1] =             [0, 1, 1]
3: [1, 1] *       [0, 1, 3, 1] =          [0, 1, 4, 4, 1]
4: [1, 3, 1] *    [0, 1, 7, 6, 1] =       [0, 1, 10, 28, 26, 9, 1]
5: [1, 7, 7, 1] * [0, 1, 15, 25, 10, 1] = [0, 1, 22, 137, 291, 261, 102, 17, 1]

Peter Luschny, Illustrating the polynomials.

Cf. A048993 (Stirling2), A205497 (zig-zag Eulerian), A320956 (row sums).

EZP(Stirling2, 7);  # Using function EZP from A373432.

A373388 Alternating row sums of the Eulerian zig-zag number triangle A205497.

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 5, 0, -45, 0, 665, 0, -14457, 0, 433741, 0, -17160421, 0, 865407905, 0, -54179057649, 0, 4122477869077, 0, -374673778941981, 0, 40087507726395689, 0, -4987405802167886825, 0, 713925031978621041757, 0, -116506260029721326349781, 0, 21501227314690679723073329

Offset: 0

Peter Luschny, Jun 03 2024

sign

Cf. A000111, A205497, A373389.

# Using the recurrence by Kyle Petersen from A205497.
G := proc(n) option remember; local F;
if n = 0 then 1/(1 - q*x) else F := G(n - 1);
simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
A373388 := n -> subs({p = 1, q = 1, x = -1}, G(n)*(1 - x)^(n + 1)):
seq(A373388(n), n = 0..20);

G[n_] := G[n] = Module[{F}, If[n == 0, 1/(1-q*x), F = G[n-1]; Simplify[ (p/(p-q))*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]]; a[n_] := a[n] = ReplaceAll[G[n]*(1-x)^(n+1), {p -> 1, q -> 1, x -> -1}]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Jun 08 2024, after Maple program *)

A373389 The Eulerian zig-zag polynomials A205497 evaluated at x = -1/2 and normalized by (-2)^n.

Original entry on oeis.org

1, 1, 1, -1, -1, 7, 1, -103, 197, 2143, -11717, -50245, 700541, 445297, -46679363, 145710035, 3366506123, -28627646249, -232615347479, 4589590326917, 7797081908429, -722997025420733, 2790363142173367, 112843029882305495, -1235970846163474579, -16017081358111849247

Offset: 0

Peter Luschny, Jun 03 2024

sign

Cf. A000111, A205497, A350354, A373388.

# Using the recurrence by Kyle Petersen from A205497.
G := proc(n) option remember; local F;
if n = 0 then 1/(1 - q*x) else F := G(n - 1);
simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
 A373389 := n -> (-2)^n*subs({p = 1, q = 1, x = -1/2}, G(n)*(1 - x)^(n + 1)):
seq(A373389(n), n = 0..22);

G[n_] := G[n] = Module[{F}, If[n == 0, 1/(1-q*x), F = G[n-1]; Simplify[ (p/(p-q))*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]];
a[n_] := a[n] = (-2)^n*ReplaceAll[G[n]*(1-x)^(n+1), {p -> 1, q -> 1, x -> -1/2}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 08 2024, after Maple program *)

A373426 Triangle read by rows: Coefficients of the polynomials L(n, x) * EZ(n, x), where L denote the unsigned Lah polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 12, 7, 1, 0, 24, 108, 144, 73, 15, 1, 0, 120, 1080, 2640, 2660, 1221, 267, 27, 1, 0, 720, 11880, 48720, 82980, 67350, 28321, 6344, 751, 44, 1, 0, 5040, 146160, 955080, 2529240, 3262350, 2245782, 870283, 195074, 25267, 1831, 68, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
  0: [1] *       [1] =                [1]
  1: [1] *       [0,  1] =            [0,  1]
  2: [1] *       [0,  2,  1] =        [0,  2,   1]
  3: [1, 1] *    [0,  6,  6,  1] =    [0,  6,  12,   7,  1]
  4: [1, 3, 1] * [0, 24, 36, 12, 1] = [0, 24, 108, 144, 73, 15, 1]

Peter Luschny, Illustrating the polynomials.

Cf. A271703 (Lah), A205497 (zig-zag Eulerian), A373425 (row sums).

# Using function EZP from A373432.
EZP((n, k) -> ifelse(n=k, 1, binomial(n-1, k-1)*n!/k!), 7);

A373429 Triangle read by rows: Coefficients of the polynomials S1(n, x) * EZ(n, x), where S1 denote the Stirling1 polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 2, -1, -2, 1, 0, -6, -7, 21, -6, -3, 1, 0, 24, 118, -147, -91, 126, -28, -3, 1, 0, -120, -1406, -109, 3749, -2084, -450, 514, -94, -1, 1, 0, 720, 16956, 34240, -72307, -15475, 56286, -21125, -674, 1635, -262, 5, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
0: [1] *          [1] =                      [1]
1: [1] *          [0,  1] =                  [0,  1]
2: [1] *          [0, -1,  1] =              [0, -1,   1]
3: [1, 1] *       [0,  2,  -3,  1] =         [0,  2,  -1,   -2,   1]
4: [1, 3, 1] *    [0, -6,  11, -6,   1] =    [0, -6,  -7,   21,  -6,  -3,  1]
5: [1, 7, 7, 1] * [0, 24, -50, 35, -10, 1] = [0, 24, 118, -147, -91, 126,-28,-3,1]

Peter Luschny, Illustrating the polynomials.

Cf. A048994 (Stirling1), A205497 (zig-zag Eulerian), A320956 (row sums).

Cf. A373428, A373432.

EZP(Stirling1, 7);  # Using function EZP from A373432.

A373431 Triangle read by rows: Coefficients of the polynomials N(n, x) * EZ(n, x), where N denote the Narayana polynomials A131198 and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 1, 1, 9, 25, 25, 9, 1, 1, 17, 97, 221, 221, 97, 17, 1, 1, 29, 291, 1229, 2476, 2476, 1229, 291, 29, 1, 1, 47, 760, 5303, 18415, 33818, 33818, 18415, 5303, 760, 47, 1, 1, 74, 1818, 19481, 106272, 317902, 544727, 544727, 317902, 106272, 19481, 1818, 74, 1

Offset: 0

Peter Luschny, Jun 05 2024

There are various conventions for indexing the Narayana, the Eulerian numbers and the zig-zag Eulerian numbers. The one we use here requires that all corresponding polynomials have p(n, 0) = 1.

			Triangle starts:
  [0] 1;
  [1] 1;
  [2] 1,  1;
  [3] 1,  4,   4,    1;
  [4] 1,  9,  25,   25,    9,    1;
  [5] 1, 17,  97,  221,  221,   97,   17,   1;
  [6] 1, 29, 291, 1229, 2476, 2476, 1229, 291, 29, 1;

Peter Luschny, Illustrating the polynomials.

Cf. A131198 (Narayana), A205497 (Eulerian zig-zag), A373430 (row sums).

R := proc(n) option remember; local F; if n = 0 then 1/(1 - q*x) else F := R(n-1);
simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1 - x)^(n + 1)) / x^2)):
nc := (n, k) -> `if`(n = 0, 0^n, binomial(n, k)^2*(n-k)/(n*(k+1))):
N := (n, x) -> local k; simplify(add(nc(n, k)*x^k, k = 0..n)):
NEZ := (n, x) -> expand(EZ(n, x) * N(n, x)):
Trow := n -> local k; if n < 2 then 1 elif n = 2 then 1, 1
else seq(coeff(NEZ(n, x), x, k), k = 0..2*n-3) fi: seq(print(Trow(n)), n = 0..6);

A205493 Third row or column of table A205497.

Original entry on oeis.org

1, 14, 109, 623, 2951, 12331, 47191, 169416, 579889, 1914226, 6144668, 19298724, 59579803, 181448918, 546629054, 1632497850, 4841448042, 14277423006, 41912838982, 122587133760, 357476552161, 1039922075888, 3019280091491, 8752184436454, 25337900299765

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

L. E. Jeffery, Unit-primitive matrices

Cf. A205497, A050446, A050447.

Python
```
# See program in A205497.
```

Conjecture 1. a(n) = M_{n,3} = M_{3,n}, where M = A205497.

Conjecture 2. Let w=2*cos(Pi/9). Then lim_{n->oo} a(n+1)/a(n) = w^3-2*w = spectral radius of the 4 X 4 unit-primitive matrix (see [Jeffery]) A_{9,3} = [0,0,0,1; 0,0,1,1; 0,1,1,1; 1,1,1,1].

a(24) and changed title from Hugo Pfoertner, Jan 05 2020

A205494 Conjectured row or column n=4 of array A205497.

Original entry on oeis.org

1, 26, 334, 2951, 20641, 123216, 656683, 3217526, 14786816, 64657546, 271838823, 1107586989, 4399926007, 17122243560, 65514790830, 247212893755, 922136438698, 3406871213836, 12486569116765, 45459575562313, 164578100859837, 593025025473647, 2128399709975819, 7613495897772440

Offset: 0

L. Edson Jeffery, Jan 28 2012

nonn

The denominator of the generating function for this sequence is a polynomial of degree 35. Terms corresponding to n=0,...,23 are shown above, with those for n=24,...,40 as follows: {27157723973468595, 96643368020414337, 343226612286408932, 1216901732483780905, 4308339945395597755, 15234940157670046379, 53818220864065451564, 189952299613455045068, 669953408386151161398, 2361449534293944339096, 8319329987059336296021, 29296032314800671782284, 103126374236214419873734, 362907786820798388773987, 1276761054260676178577043, 4490840947292979020061377, 15793032895427304036405557}.

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

L. E. Jeffery, Unit-primitive matrices

Cf. A205497, A050446, A050447.

G.f.: (1+4*x-31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28) / ((1-x)^5 * (1-x-x^2)^4 * (1-2*x-x^2+x^3)^3 * (1-2*x-3*x^2+x^3+x^4)^2 * (1-3*x-3*x^2+4*x^3+x^4-x^5)).
CONJECTURE 1. a(n) = M_{n,4} = M_{4,n}, where M = A205497.
CONJECTURE 2. Let w=2*cos(Pi/11). Then lim_{n->oo} a(n+1)/a(n) = w^4-3*w^2+1 = spectral radius of the 5 X 5 unit-primitive matrix (see [Jeffery]) A_{11,4} = [0,0,0,0,1; 0,0,0,1,1; 0,0,1,1,1; 0,1,1,1,1; 1,1,1,1,1].

A373427 Triangle read by rows: Coefficients of the polynomials SC(n, x) * EZ(n, x), where SC denote the Stirling cycle polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 6, 29, 45, 30, 9, 1, 0, 24, 218, 553, 629, 366, 112, 17, 1, 0, 120, 1954, 7781, 13409, 12136, 6270, 1894, 326, 29, 1, 0, 720, 20484, 125968, 313715, 407297, 308286, 143725, 42124, 7683, 830, 47, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
0: [1] *          [1] =                    [1]
1: [1] *          [0,  1] =                [0,  1]
2: [1] *          [0,  1, 1] =             [0,  1,   1]
3: [1, 1] *       [0,  2, 3, 1] =          [0,  2,   5,   4,   1]
4: [1, 3, 1] *    [0,  6, 11, 6, 1] =      [0,  6,  29,  45,  30,   9,   1]
5: [1, 7, 7, 1] * [0, 24, 50, 35, 10, 1] = [0, 24, 218, 553, 629, 366, 112,17,1]

Peter Luschny, Illustrating the polynomials.

Cf. A132393 (Stirling cycle), A205497 (zig-zag Eulerian), A373433 (row sums).

Cf. A373429, A373428, A373432.

EZP((n, k) -> abs(Stirling1(n, k)), 7);  # Using function EZP from A373432.

cp's OEIS Frontend

A373432 Triangle read by rows. Coefficients of the polynomials P(n, x) * EZ(n, x), where P denote the Pascal polynomials and EZ the zig-zag Eulerian polynomials A205497.

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A373428 Triangle read by rows: Coefficients of the polynomials S2(n, x) * EZ(n, x), where S2 denote the Stirling set polynomials and EZ the Eulerian zig-zag polynomials A205497.

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A373388 Alternating row sums of the Eulerian zig-zag number triangle A205497.

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Mathematica

A373389 The Eulerian zig-zag polynomials A205497 evaluated at x = -1/2 and normalized by (-2)^n.

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A373426 Triangle read by rows: Coefficients of the polynomials L(n, x) * EZ(n, x), where L denote the unsigned Lah polynomials and EZ the Eulerian zig-zag polynomials A205497.

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A373429 Triangle read by rows: Coefficients of the polynomials S1(n, x) * EZ(n, x), where S1 denote the Stirling1 polynomials and EZ the Eulerian zig-zag polynomials A205497.

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A373431 Triangle read by rows: Coefficients of the polynomials N(n, x) * EZ(n, x), where N denote the Narayana polynomials A131198 and EZ the Eulerian zig-zag polynomials A205497.

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Maple

A205493 Third row or column of table A205497.

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A205494 Conjectured row or column n=4 of array A205497.

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