Real-rooted atlas

A038336 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(72 + 240\,x + 150\,x^{2} + 60\,n x + 50\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 480\,x + 300\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0144480300924
11728864010800400025168
22073613824025920019200050000660176
3248832207360051840005760000300000060000016866432
429859842985984093312000138240000108000000432000007000000422597824
53583180841803776015676416002903040000302400000018144000005880000008000000010430951168
6429981696573308928025082265600557383680007257600000058060800000282240000007680000000900000000254424504576
7515978035277396705280386983526400100329062400015676416000001567641600000101606400000041472000000097200000000100000000006146097836032

All 15 rows

Technical details
Node id
n-f0c64936dd0c4060 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7d1144c6b7b8d949

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          144,            480,            300;
  1:         1728,           8640,          10800,            4000;
  2:        20736,         138240,         259200,          192000,           50000;
  3:       248832,        2073600,        5184000,         5760000,         3000000,          600000;
  4:      2985984,       29859840,       93312000,       138240000,       108000000,        43200000,         7000000;
  5:     35831808,      418037760,     1567641600,      2903040000,      3024000000,      1814400000,       588000000,        80000000;
  6:    429981696,     5733089280,    25082265600,     55738368000,     72576000000,     58060800000,     28224000000,      7680000000,       900000000;
  7:   5159780352,    77396705280,   386983526400,   1003290624000,   1567641600000,   1567641600000,   1016064000000,    414720000000,     97200000000,    10000000000;
  8:  61917364224,  1031956070400,  5804752896000,  17199267840000,  31352832000000,  37623398400000,  30481920000000,  16588800000000,   5832000000000,  1200000000000,   110000000000;
  9: 743008370688, 13621820129280, 85136375808000, 283787919360000, 591224832000000, 827714764800000, 804722688000000, 547430400000000, 256608000000000, 79200000000000, 14520000000000, 1200000000000;
%F Formula
The row polynomials satisfy (6+(5*n+10)*x)*P(n,x) = (72+(60*n+240)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 300*x^2.
%t Mathematica
P[0, x_] := 144 + 480*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (60*n + 240)*x + (50*n + 150)*x^2)*P[n - 1, x])/(6 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 480*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (60*n + 240)*x + (50*n + 150)*x^2)*v[i - 1])/(6 + (5*n + 10)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.