Real-rooted atlas

A145324 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 6\,n x - 3\,n^{2} - 12\,n^{2} x - 9\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 + 4\,n + 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
251108159
3399176419444107
43294225545135438880116082
52953827646296843615046568748003653892
6289224345229216456041391534564649562022044960127891593
7308685644953380272386881878974416159306426015374784966172588804947203169
83576744061622402445444807001859567187645596029140669900669845460316142419046845440210028247028
944787384089245106647771936878038633161863611988940158902378347785864294528712010420816236915206428310336009720407018898
1060331132801366538014081374395703228805563164995230322907459415760163764596121269098351970121360588842564488499425195088023570471232000487445440823235
11870525014402210883574464252464494179961706942052043207567012212202352298162496221684484251566886486069901987148680326615310087747584370618620405120093339066078720026340703644459015
1213401619257603774246515481648278698095808837025342425962721891515960079389067587892100210574172361612942194662313700623248986208399309309843301080338550043992445440172062857404817280397115299316736001526423432915057670
132193317300160067871057908608096195800191247288257408457259700847811044174821263019671424253831706665897230907043238056129788667699601368962081199409616411452823712685119240732000182219548935232385288479280233704913920180687461189114880094447033807002715440
143803534413056001283335483552128019972624439430604818983601543090358081229455932517998378057292242660534212676197630671845308268861510948029253014433216990047035767857762808142000266671605743840014653291916987880504481030200814875873968640442274905183372934400875639234993402880006215695340932307509965
Technical details
Node id
n-63a4f3580f22ae58 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-50c545bfd2603dc2

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:         6;
  2:        51,        108;
  3:       399,       1764,        1944;
  4:      3294,      22554,       51354,        38880;
  5:     29538,     276462,      968436,      1504656,        874800;
  6:    289224,    3452292,    16456041,     39153456,      46495620,      22044960;
  7:   3086856,   44953380,   272386881,    878974416,    1593064260,    1537478496,     617258880;
  8:  35767440,  616224024,  4544480700,  18595671876,   45596029140,   66990066984,   54603161424,   19046845440;
  9: 447873840, 8924510664, 77719368780, 386331618636, 1198894015890, 2378347785864, 2945287120104, 2081623691520, 642831033600;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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.