Real-rooted atlas

A094816 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 36\,x - 9\,n + 12\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(36\,x^{2} - 12\,n x^{2}\right)\,P_{n-3}'(x) + \left(4 + 12\,x + 81\,x^{2} - 12\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-12\,x^{2} + 36\,x^{3} + 12\,n x^{2}\right)\,P_{n-2}'(x) + \left(-1 + 12\,x + 2\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 12\,x\)
\(P_{3}(x) = 5 + 60\,x + 81\,x^{2}\)
\(P_{4}(x) = 16 + 276\,x + 729\,x^{2} + 432\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
221214
356081146
4162767294321453
565139252656048202514795
632680283807062640405008748158312
7195753148294354611280567000236196357211799656
8137004005002485161608428871583754286520125023513996821818747
9109601339686423009346637338248897647567849488285053586158592531441282270989
1098641032079852233141004711380448113125005010285810925521037761721606402869781419683003892349386
119864101334143516257400593184949992001497755812515590746836100166327734001685120964565415127939500714492957099285446
121085051123807292884308121298651085542175522082385057502415062022121783407717638554746182426481705030508805550055015953325509168889060516193
13130206134547121290160397942849971148209119625630511953301503867672758400319439739811517669031240966584649649381628177760002554312117522958251208981352914657837509255
1416926797486629635283772551952304979421568988003472471651750355506443050801730458374471409254362268966533761560943934807446571513274009420303089341230307172649340607016312487308255273429046004
15236975164805903507405433281869626862748333705090766320768916717507575111993143055393610971811460339737491663318522243631305439077031255533771492003070385337719751788967965285721121797303718598965210761680254685219438363948
Technical details
Node id
n-784751e4fe3ae02f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cc7836baf34fb4ee

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      2,       12;
   3:      5,       60,        81;
   4:     16,      276,       729,       432;
   5:     65,     1392,      5265,      6048,       2025;
   6:    326,     8028,     38070,     62640,      40500,       8748;
   7:   1957,    53148,    294354,    611280,     567000,     236196,     35721;
   8:  13700,   400500,   2485161,   6084288,    7158375,    4286520,   1250235,    139968;
   9: 109601,  3396864,  23009346,  63733824,   88976475,   67849488,  28505358,   6158592,   531441;
  10: 986410, 32079852, 233141004, 711380448, 1131250050, 1028581092, 552103776, 172160640, 28697814, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(12*n-36)*x)*P(n-3,x) - (12*n-36)*x^2*d/dx P(n-3,x) - (n^2-4+(12*n-12)*x-81*x^2)*P(n-2,x) + ((12*n-12)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n-1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 12*x, P(3,x) = 5 + 60*x + 81*x^2, P(4,x) = 16 + 276*x + 729*x^2 + 432*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 12*x;
P[3, x_] := 5 + 60*x + 81*x^2;
P[4, x_] := 16 + 276*x + 729*x^2 + 432*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (12*n - 36)*x)*P[n - 3, x] - (12*n - 36)*x^2*D[P[n - 3, x], x] - (n^2 - 4 + (12*n - 12)*x - 81*x^2)*P[n - 2, x] + ((12*n - 12)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 12*x; v[3] = 5 + 60*x + 81*x^2; v[4] = 16 + 276*x + 729*x^2 + 432*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (12*n - 36)*x)*v[i - 3] - (12*n - 36)*x^2*deriv(v[i - 3], x) - (n^2 - 4 + (12*n - 12)*x - 81*x^2)*v[i - 2] + ((12*n - 12)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 12*x)*v[i - 1] - 12*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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023