Real-rooted atlas

A089231 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-1 - 2\,x + n + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 2\,x - x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
2181230
315621672444
41460312021604807220
5150304380046800216003600130830
6170142631260919800655200226800302402633442
72107448952795217675280171696009172800254016028224058475480
8283934161517362563430062724242067203090528001320883203048192029030401421868744
941368977025554074406828131520102901881609544651200556295040019813248003919104003265920037600912890
106483520010455058747001405474092002503648224002829801744002099823264001019874240003113510400053887680003991680001074774591110
1110877765461285582464132030033877302006184086004800826203913920074706766041604619611180800192319142400051372921600079035264000526901760033025627876692
12194534530834816969314119472667543220229601561761619704002411793541872002577756211430401942375917081601029513348864003750223276800089046397440001232950118400747242496001085703592225980
1336941705020766354052846119336154420758487195240497622027262407106015369653200877892849241408078191938413388805050177384412160234214286866560075837848486400016206444334080020399720140800113331778560038023398261353414
1474235368012463077577580543608603717554884253028010809453094103664021261251564312760029845264552543440030726249723449280023457581524016640013256715634081920054650000268864000159259481821440003093957554688000356995102464000183074411520001413286030051404810
Technical details
Node id
n-471b3d369e427d51 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-01d09cfb9e6b0947

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:         2;
  2:        18,         12;
  3:       156,        216,         72;
  4:      1460,       3120,       2160,         480;
  5:     15030,      43800,      46800,       21600,       3600;
  6:    170142,     631260,     919800,      655200,     226800,      30240;
  7:   2107448,    9527952,   17675280,    17169600,    9172800,    2540160,     282240;
  8:  28393416,  151736256,  343006272,   424206720,  309052800,  132088320,   30481920,   2903040;
  9: 413689770, 2555407440, 6828131520, 10290188160, 9544651200, 5562950400, 1981324800, 391910400, 32659200;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n^2+n-1+(2*n^2-2)*x)*P(n-1,x) - (n+1+(2*n+2)*x+(n+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n - 1 + (2*n^2 - 2)*x)*P[n - 1, x] - (n + 1 + (2*n + 2)*x + (n + 1)*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] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n - 1 + (2*n^2 - 2)*x)*v[i - 1] - (n + 1 + (2*n + 2)*x + (n + 1)*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

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.