Real-rooted atlas

A039756 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,x - x^{3} + 2\,n x^{3}\right)\,P_{n}(x) = \left(-16 - 44\,x - 16\,x^{2} + 28\,x^{3} + 12\,x^{4} - 3\,x^{5} + 16\,n + 24\,n x + 16\,n x^{3} + 28\,n x^{4} + 2\,n x^{5} + 8\,n^{2} x - 20\,n^{2} x^{3} + 12\,n^{2} x^{5} - 8\,n^{3} x^{3} - 16\,n^{3} x^{4} - 8\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(-8 - 24\,x - 16\,x^{2} + 12\,x^{3} + 14\,x^{4} - 2\,x^{6} - 8\,n x - 16\,n x^{2} + 4\,n x^{3} + 24\,n x^{4} + 12\,n x^{5} + 8\,n^{2} x^{3} + 24\,n^{2} x^{4} + 24\,n^{2} x^{5} + 8\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-8 - 4\,x + 8\,x^{2} + 2\,x^{3} + 2\,x^{4} - 8\,n x + 4\,n x^{3} - 8\,n x^{4} + 8\,n^{2} x^{3} + 8\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(4\,x + 8\,x^{2} + 6\,x^{3} + 4\,x^{4} + 2\,x^{5} - 4\,n x^{3} - 8\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2612321
32476484152
411651255217651361
5648368856883296630613956
640882859257708490561803022207159701
72864023832059726467521637468094788770082016616
82199202126336638688090627846806280268896049123226304927808705
918322242021430470928832121522432115817840630772321869022425438728865010414715620
101643017620390982481975979216495219201912367280133655318455638760812791001613245318295340116636380469
11157554048217441811298652184322283729056031264796480267831919681447275961647670060168700536886945014097442412113262713496
121606879040244285992961235698579203238716561925122175298405210364341763478159211521505011427844018041414059159657603663847443188784132051513973841
13173502557442882535153921609621665408471596701747284721568481609997921024800797013044128042950937640961525530511800335982943000403592427921941063840103632621439270318657060
141975536454403562673738496217800885567367059539952384014214400784656019117005979756817755772348014411493437361920051270700576104152237049636002802775434756276911246496103068302223348097215791526312740149
Technical details
Node id
n-2b4dc9c4a986411b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-479f84db24b17c92

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:       2,        2;
  2:       6,       12,        3;
  3:      24,       76,       48,         4;
  4:     116,      512,      552,       176,         5;
  5:     648,     3688,     5688,      3296,       630,        6;
  6:    4088,    28592,    57708,     49056,     18030,     2220,        7;
  7:   28640,   238320,   597264,    675216,    374680,    94788,     7700,       8;
  8:  219920,  2126336,  6386880,   9062784,   6806280,  2688960,   491232,   26304,     9;
  9: 1832224, 20214304, 70928832, 121522432, 115817840, 63077232, 18690224, 2543872, 88650, 10;
%F Formula
The row polynomials satisfy -(2*x-(2*n-1)*x^3)*P(n,x) = (16*n-16+(8*n^2+24*n-44)*x-16*x^2-(8*n^3+20*n^2-16*n-28)*x^3-(16*n^3-28*n-12)*x^4-(8*n^3-12*n^2-2*n+3)*x^5)*P(n-2,x) - (8+(8*n+24)*x+(16*n+16)*x^2-(8*n^2+4*n+12)*x^3-(24*n^2+24*n+14)*x^4-(24*n^2+12*n)*x^5-(8*n^2-2)*x^6)*d/dx P(n-2,x) - (8+(8*n+4)*x-8*x^2-(8*n^2+4*n+2)*x^3-(8*n^2-8*n+2)*x^4)*P(n-1,x) + (4*x+8*x^2-(4*n-6)*x^3-(8*n-4)*x^4-(4*n-2)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 16 + (8*n^2 + 24*n - 44)*x - 16*x^2 - (8*n^3 + 20*n^2 - 16*n - 28)*x^3 - (16*n^3 - 28*n - 12)*x^4 - (8*n^3 - 12*n^2 - 2*n + 3)*x^5)*P[n - 2, x] - (8 + (8*n + 24)*x + (16*n + 16)*x^2 - (8*n^2 + 4*n + 12)*x^3 - (24*n^2 + 24*n + 14)*x^4 - (24*n^2 + 12*n)*x^5 - (8*n^2 - 2)*x^6)*D[P[n - 2, x], x] - (8 + (8*n + 4)*x - 8*x^2 - (8*n^2 + 4*n + 2)*x^3 - (8*n^2 - 8*n + 2)*x^4)*P[n - 1, x] + (4*x + 8*x^2 - (4*n - 6)*x^3 - (8*n - 4)*x^4 - (4*n - 2)*x^5)*D[P[n - 1, x], x])/(-2*x + (2*n - 1)*x^3)]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((16*n - 16 + (8*n^2 + 24*n - 44)*x - 16*x^2 - (8*n^3 + 20*n^2 - 16*n - 28)*x^3 - (16*n^3 - 28*n - 12)*x^4 - (8*n^3 - 12*n^2 - 2*n + 3)*x^5)*v[i - 2] - (8 + (8*n + 24)*x + (16*n + 16)*x^2 - (8*n^2 + 4*n + 12)*x^3 - (24*n^2 + 24*n + 14)*x^4 - (24*n^2 + 12*n)*x^5 - (8*n^2 - 2)*x^6)*deriv(v[i - 2], x) - (8 + (8*n + 4)*x - 8*x^2 - (8*n^2 + 4*n + 2)*x^3 - (8*n^2 - 8*n + 2)*x^4)*v[i - 1] + (4*x + 8*x^2 - (4*n - 6)*x^3 - (8*n - 4)*x^4 - (4*n - 2)*x^5)*deriv(v[i - 1], x))/(-2*x + (2*n - 1)*x^3)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.