Real-rooted atlas

A008277 → θ+1 → D+1 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(4\right)\,P_{n}(x) = \left(16 + 6\,x - 6\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(3 - x - 9\,x^{2}\right)\,P_{n-3}'(x) + \left(x - 3\,x^{2} - 4\,x^{3}\right)\,P_{n-3}''(x) + \left(-30 - 12\,x - 12\,x^{2}\right)\,P_{n-2}(x) + \left(-4 + 4\,x\right)\,P_{n-2}'(x) + \left(-x + 4\,x^{2}\right)\,P_{n-2}''(x) + \left(18 + 12\,x\right)\,P_{n-1}(x) + \left(1\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 4\,x\)
\(P_{3}(x) = 7 + 24\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2347
3724940
4151009016221
531360585240251241
663120431502240525367218
712738641530916800665010084943807
825512100699301112166615016632176464278111
951137320306945680400570675211680367502880811847242
101023114004131175039476804487175230353258212073920445510012825759
112047346104550350922070400330913002258625677923231425600138105660012192962365
124095104650022792770120136016233032800205755264930809882300812831853252431009438144702294568
138191315588093525705641200560158590932517758540801024703680328648320609729126606600407407131041695521005933
14163839500404381203550337227072010517331825147146879881061476416042962233601025674650147947800128828706552001774519645093176851
153276728566744154619010917539048800683731639501181477737441049464385975253019200015694154190289369080033361127823849280101738023520225382057753384
Technical details
Node id
n-d15b68a7dc3afe9f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-44fef67f8d84e135

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    3,      4;
   3:    7,     24,       9;
   4:   15,    100,      90,      16;
   5:   31,    360,     585,     240,      25;
   6:   63,   1204,    3150,    2240,     525,      36;
   7:  127,   3864,   15309,   16800,    6650,    1008,     49;
   8:  255,  12100,   69930,  111216,   66150,   16632,   1764,    64;
   9:  511,  37320,  306945,  680400,  570675,  211680,  36750,  2880,   81;
  10: 1023, 114004, 1311750, 3947680, 4487175, 2303532, 582120, 73920, 4455, 100;
%F Formula
The row polynomials satisfy 4*P(n,x) = (16+6*x-6*x^2+4*x^3)*P(n-3,x) + (3-x-9*x^2)*d/dx P(n-3,x) + (x-3*x^2-4*x^3)*d^2/dx^2 P(n-3,x) - (30+12*x+12*x^2)*P(n-2,x) - (4-4*x)*d/dx P(n-2,x) - (x-4*x^2)*d^2/dx^2 P(n-2,x) + (18+12*x)*P(n-1,x) + d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 4*x, P(3,x) = 7 + 24*x + 9*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 4*x;
P[3, x_] := 7 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 6*x - 6*x^2 + 4*x^3)*P[n - 3, x] + (3 - x - 9*x^2)*D[P[n - 3, x], x] + (x - 3*x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] - (30 + 12*x + 12*x^2)*P[n - 2, x] - (4 - 4*x)*D[P[n - 2, x], x] - (x - 4*x^2)*D[P[n - 2, x], {x, 2}] + (18 + 12*x)*P[n - 1, x] + D[P[n - 1, x], x])/(4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 4*x; v[3] = 7 + 24*x + 9*x^2; for(n = 4, N, my(i = n); v[i] = ((16 + 6*x - 6*x^2 + 4*x^3)*v[i - 3] + (3 - x - 9*x^2)*deriv(v[i - 3], x) + (x - 3*x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) - (30 + 12*x + 12*x^2)*v[i - 2] - (4 - 4*x)*deriv(v[i - 2], x) - (x - 4*x^2)*deriv(deriv(v[i - 2], x), x) + (18 + 12*x)*v[i - 1] + deriv(v[i - 1], x))/(4)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.