Real-rooted atlas

A059364 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 4\,n x - 4\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
215442483
31054885641921349
49456072109327744192027613
5103958569221133223094411792023040679323
61351351363476428977263835844997120199756832256019489215
720270252423592092979576175829504186669520113915136374653445160960638282985
834459425476875440216231487249726658566689971920552061209627633267207728168969289728023485940505
9654729075103011726605395961944814637620819223899444824024939502329616843606169671529099264174145904641857945600958918897935
1013749310575242513788860144218718820845128235573128678288617840109731622469769356374505856535207820902419751111331844260077568004087480320043013171117835
1131623414322561827868090804117326036116414609909933894432370328603640048148955119142449684111130707235834378638336017789149173657658078672773120112495081881609809952768002102349783545325
1279058535806251697489091018001251911834155980497002181503084812475555131078320213598934176339202590273521603667222586236569720832141225695257651206195739932303360181451466591436831905745993728025505877196800111201396207455925
132134580466768754994324931941100404218269771816601775901992565735364983656975276172009659544423657594241347290617834552384137950966237326643210419567968599925765752703200276275202261900580435640326008078557878681696754438058803207141645615104006328227798190930275
14619028335362937515678907364019690013819946294797943406659197318922081184206682622022809725604475776626553496121670625939701150160320831190799795304028167367130147369767731249099272960889856000242778375401142845448651420918388228096210303836626039603231246261627650048021424936845312000385511978266364453175
Technical details
Node id
n-73921d4b844384c9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-aaef48f0a1fd409f

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:         3,           4;
  2:        15,          44,          24;
  3:       105,         488,         564,          192;
  4:       945,        6072,       10932,         7744,         1920;
  5:     10395,       85692,      211332,       230944,       117920,        23040;
  6:    135135,     1363476,     4289772,      6383584,      4997120,      1997568,       322560;
  7:   2027025,    24235920,    92979576,    175829504,    186669520,    113915136,     37465344,     5160960;
  8:  34459425,   476875440,  2162314872,   4972665856,   6689971920,   5520612096,   2763326720,   772816896,    92897280;
  9: 654729075, 10301172660, 53959619448, 146376208192, 238994448240, 249395023296, 168436061696, 71529099264, 17414590464, 1857945600;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 1)); 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.