Real-rooted atlas

A269944 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 - 40\,x + 36\,x^{2} + 40\,n + 32\,n x - 26\,n^{2} - 8\,n^{2} x + 8\,n^{3} - n^{4}\right)\,P_{n-2}(x) + \left(40\,x^{2} + 16\,x^{3} - 32\,n x^{2} + 8\,n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(7 + 8\,x - 6\,n + 2\,n^{2}\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1189
22243662
310136288128562
41001440349223044007736
51700252806584451584140001152159560
644200670880182570415752965252007027231364714688
716354002517616070570008647773442435520041126403073288192190942272

All 15 rows

Technical details
Node id
n-70b96780de4faf54 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8ef17fcdd7f978df

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          1,           8;
  2:          2,          24,           36;
  3:         10,         136,          288,          128;
  4:        100,        1440,         3492,         2304,          400;
  5:       1700,       25280,        65844,        51584,        14000,        1152;
  6:      44200,      670880,      1825704,      1575296,       525200,       70272,       3136;
  7:    1635400,    25176160,     70570008,     64777344,     24355200,     4112640,     307328,      8192;
  8:   81770000,  1271891200,   3641793120,   3489782784,   1420189200,   275774976,   26561920,   1212416,   20736;
  9: 5315050000, 83327088000, 242440063200, 239784478720, 103217869200, 22015518336, 2477245568, 148193280, 4416768, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(8*n^2-32*n+40)*x-36*x^2)*P(n-2,x) + ((8*n^2-32*n+40)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+7+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*P[n - 2, x] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 7 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*v[i - 2] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 7 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.