Real-rooted atlas

A200536 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 14\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 14 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1141630
21685846722561680
315128784203282342413440307270560
41209698496342720660544761600525312200704327682634240
5907209504004415040119366402069872023873280183142409011200258048032768092198400
66531848366976486259201692403203919507206342178567316413446017433603457843201322188803027763231457283097866240
74572288692375044831384322056997376596175552012420963072191385877762208171212819077617664121896140805598838784175059763233397145629360128101196963840

All 15 rows

Technical details
Node id
n-3fbcf027e0e0a748 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-cc36510735305ef0

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         0;
  1:        14,         16;
  2:       168,        584,         672,          256;
  3:      1512,       8784,       20328,        23424,        13440,          3072;
  4:     12096,      98496,      342720,       660544,       761600,        525312,        200704,          32768;
  5:     90720,     950400,     4415040,     11936640,     20698720,      23873280,      18314240,        9011200,        2580480,         327680;
  6:    653184,    8366976,    48625920,    169240320,    391950720,     634217856,     731641344,      601743360,      345784320,      132218880,       30277632,        3145728;
  7:   4572288,   69237504,   483138432,   2056997376,   5961755520,   12420963072,   19138587776,    22081712128,    19077617664,    12189614080,     5598838784,     1750597632,     333971456,      29360128;
  8:  31352832,  547928064,  4462553088,  22468368384,  78209832960,  199367239680,  384478918656,   571197949952,   659106717696,   590717747200,   407849795584,   213033418752,   81489035264,   21550333952,    3523215360,    268435456;
  9: 211631616, 4192321536, 39034275840, 226878050304, 921956152320, 2779749052416, 6439155222528, 11705434288128, 16905648324096, 19509057146880, 17988751097856, 13178069581824, 7575975493632, 3346217828352, 1096600780800, 251255586816, 35936796672, 2415919104;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*n+14*n*x+8*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 14 + 16*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 14 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 14*n*x + 8*n*x^2)*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] = 0; v[2] = 14 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 14*n*x + 8*n*x^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.