Real-rooted atlas

A059364 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 8\,n x + 36\,n x^{2} + 4\,n^{2} + 24\,n^{2} x - 72\,n^{2} x^{2} - 4\,n^{3} - 16\,n^{3} x + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,n x^{2} + 16\,n x^{3} - 24\,n^{2} x^{2} - 32\,n^{2} x^{3} + 16\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 2\,n - 8\,n x + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2158872175
310597616927683541
4945121443279630976960086461
5103951713846339969237765896001382402467391
6135135272695212869316255343362498560011985408225792080494667
7202702548471840278938728703318016933347600683490816262257408412876802953139113
8344594259537508806486944616198906634243344985960033123672576193432870406182535168836075520120301248249
965472907520602345320161878858344585504832768119497224120014963701397761179052431872572232794112156731314176185794560005386579142643
101374931057548502757772043265615646241805129422924843391443089200658389734818566549462154099242816625672192177760001986564260077568000449622835200262903997068263
1131623414322512365573618160123519781083492584396397355776161851643018200028889373071485443477887779149504286675029106688016010234256291845807867277312001237445900697601177194332160013890016480499325
1279058535806253394978182036003755735502467940198800872601233926237777565539160012815936050580352018131914651225670418068989255776665612710312573188608061957399323033600199596613250580483828689519247360331576403558400789709853968377525
13213458046676875998864986388220012126548093154498071036079702629414424918284876380860005795726654194556544943103432484186668811036077298986131456937761117173993318457527032002762752002488090638479204352720969426945441792125780769476444160999830386114560048071648662307483175
1461902833536293753135781472803938004145983888439383020266367892756883247361033413110114048628002685465975932097672964943815779080511222406649526398362432225286630417132632790958084909927296088985600002670562129412571299841038170510206587371522733949876138514841643744766278710067203213740526796800003119267724219700063875
Technical details
Node id
n-66c5ba5b8b26ed07 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dbeaff73e6946047

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,           8;
  2:        15,          88,           72;
  3:       105,         976,         1692,          768;
  4:       945,       12144,        32796,        30976,          9600;
  5:     10395,      171384,       633996,       923776,        589600,        138240;
  6:    135135,     2726952,     12869316,     25534336,      24985600,      11985408,       2257920;
  7:   2027025,    48471840,    278938728,    703318016,     933347600,     683490816,     262257408,     41287680;
  8:  34459425,   953750880,   6486944616,  19890663424,   33449859600,   33123672576,   19343287040,   6182535168,    836075520;
  9: 654729075, 20602345320, 161878858344, 585504832768, 1194972241200, 1496370139776, 1179052431872, 572232794112, 156731314176, 18579456000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(16*n^3-24*n^2+8*n)*x-(36*n^3-72*n^2+36*n)*x^2)*P(n-2,x) + ((16*n^3-24*n^2+8*n)*x^2+(16*n^3-32*n^2+16*n)*x^3)*d/dx P(n-2,x) + (4*n^2-2*n-1+(8*n^2-8*n)*x)*P(n-1,x) - (8*n^2-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x - (36*n^3 - 72*n^2 + 36*n)*x^2)*P[n - 2, x] + ((16*n^3 - 24*n^2 + 8*n)*x^2 + (16*n^3 - 32*n^2 + 16*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x] - (8*n^2 - 8*n)*x^2*D[P[n - 1, x], 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (16*n^3 - 24*n^2 + 8*n)*x - (36*n^3 - 72*n^2 + 36*n)*x^2)*v[i - 2] + ((16*n^3 - 24*n^2 + 8*n)*x^2 + (16*n^3 - 32*n^2 + 16*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 2*n - 1 + (8*n^2 - 8*n)*x)*v[i - 1] - (8*n^2 - 8*n)*x^2*deriv(v[i - 1], x))/(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.