Real-rooted atlas

A156365 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-24\,x + 200\,x^{2} - 32\,n x + 372\,n x^{2} - 8\,n^{2} x + 208\,n^{2} x^{2} + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x + 74\,x^{2} - 200\,x^{3} - 2\,n x + 48\,n x^{2} - 84\,n x^{3} + 8\,n^{2} x^{2} + 40\,n^{2} x^{3} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-3\,x^{2} + 50\,x^{3} - 88\,x^{4} - n x^{2} + 40\,n x^{3} - 76\,n x^{4} + 8\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(2 + 32\,x + n + 32\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(5\,x - 48\,x^{2} + 2\,n x - 40\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 32\,x + 36\,x^{2}\)
\(P_{1}(x) = 1 + 88\,x + 396\,x^{2} + 128\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01323669
1188396128613
21208237633284006313
31456108723865622800115273937
41960428763092484764001382403136970861
511976154548199923262476004945536774592819214131677
614016525888112939526247600010164556845810688411238420736225889233
7181041722240582645765241416001509527808142748211239190528021005568512003934108489
811628854949322820464643895245600181143336963053872550418050973696316508083210424320012390474156284117
913266417217756130402368026527389600187214153472509638528896543280939008211251836160244874752005059000322949121504227791381
10165424532701365828281600169312614000173447597721671349394045441233405139353687771659097602331312640000183343970304241179033669222432672896009081
1111309521633433042539658060810286968700001479543881817687717608224512229139874545664266317898727168131673199360000245838900285441338108346368113310146561605632756631607595585
12126201649769060410855563673660164918084001183991027650569765115086168963666237992730624645693324065049652621823451136001863668502569984250112187039744956981612953652587659264368640018609792832359357
131524152150957882045722691264034153295419600900997077080448100586672467003525228788879749120013235371851864960016422313872163840096907241179319296251805416469626882472683143557120673285572198402415292416008388608484487058458684365
14110484324564116720190405112960018934132042400065863203521840649739052764614476868106013233455104023823669333356928004256625827090944000384791962291706675216860980101591203843276665154729574402388441736970240046736555212800010993606328321893990413310258021546783137
Technical details
Node id
n-e4a2fa0161e24749 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c7274fd18fd89701

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,    32,       36;
  1: 1,    88,      396,        128;
  2: 1,   208,     2376,       3328,         400;
  3: 1,   456,    10872,      38656,       22800,         1152;
  4: 1,   960,    42876,     309248,      476400,       138240,         3136;
  5: 1,  1976,   154548,    1999232,     6247600,      4945536,       774592,         8192;
  6: 1,  4016,   525888,   11293952,    62476000,    101645568,     45810688,      4112384,        20736;
  7: 1,  8104,  1722240,   58264576,   524141600,   1509527808,   1427482112,    391905280,     21005568,       51200;
  8: 1, 16288,  5494932,  282046464,  3895245600,  18114333696,  30538725504,  18050973696,   3165080832,   104243200,    123904;
  9: 1, 32664, 17217756, 1304023680, 26527389600, 187214153472, 509638528896, 543280939008, 211251836160, 24487475200, 505900032, 294912;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -((8*n^2+32*n+24)*x-(36*n^3+208*n^2+372*n+200)*x^2)*P(n-2,x) - ((2*n+6)*x-(8*n^2+48*n+74)*x^2-(16*n^3+40*n^2-84*n-200)*x^3)*d/dx P(n-2,x) - ((n+3)*x^2-(8*n^2+40*n+50)*x^3+(16*n^2+76*n+88)*x^4)*d^2/dx^2 P(n-2,x) + (n+2+(8*n^2+32*n+32)*x)*P(n-1,x) + ((2*n+5)*x-(8*n^2+40*n+48)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 32*x + 36*x^2, P(1,x) = 1 + 88*x + 396*x^2 + 128*x^3.
%t Mathematica
P[0, x_] := 1 + 32*x + 36*x^2;
P[1, x_] := 1 + 88*x + 396*x^2 + 128*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^2 + 32*n + 24)*x - (36*n^3 + 208*n^2 + 372*n + 200)*x^2)*P[n - 2, x] - ((2*n + 6)*x - (8*n^2 + 48*n + 74)*x^2 - (16*n^3 + 40*n^2 - 84*n - 200)*x^3)*D[P[n - 2, x], x] - ((n + 3)*x^2 - (8*n^2 + 40*n + 50)*x^3 + (16*n^2 + 76*n + 88)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 2 + (8*n^2 + 32*n + 32)*x)*P[n - 1, x] + ((2*n + 5)*x - (8*n^2 + 40*n + 48)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 32*x + 36*x^2; v[2] = 1 + 88*x + 396*x^2 + 128*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^2 + 32*n + 24)*x - (36*n^3 + 208*n^2 + 372*n + 200)*x^2)*v[i - 2] - ((2*n + 6)*x - (8*n^2 + 48*n + 74)*x^2 - (16*n^3 + 40*n^2 - 84*n - 200)*x^3)*deriv(v[i - 2], x) - ((n + 3)*x^2 - (8*n^2 + 40*n + 50)*x^3 + (16*n^2 + 76*n + 88)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 2 + (8*n^2 + 32*n + 32)*x)*v[i - 1] + ((2*n + 5)*x - (8*n^2 + 40*n + 48)*x^2)*deriv(v[i - 1], x))/(n + 2)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.