Real-rooted atlas

A038329 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(15 + 39\,x + 24\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 225 + 720\,x + 576\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02257205761521
1337516200259201382459319
2506253240007776008294403317762313441
37593756075000194400003110400024883200796262490224199
41139062510935000043740000093312000011197440007166361601911029763518743761
517085937519136250009185400000244944000003919104000037623398400200658124804586471424137231006679
62562890625328050000001837080000005878656000001175731200000150493593600012039487488005503765708801100753141765352009260481
73844335937555358437500035429400000001322697600000031744742400000507915878400005417769369600037150418534400148601674137602641807540224208728361158759

All 15 rows

Technical details
Node id
n-bb08c7dea0976170 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b28493176e02e0b

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:           225,             720,              576;
  1:          3375,           16200,            25920,            13824;
  2:         50625,          324000,           777600,           829440,            331776;
  3:        759375,         6075000,         19440000,         31104000,          24883200,           7962624;
  4:      11390625,       109350000,        437400000,        933120000,        1119744000,         716636160,         191102976;
  5:     170859375,      1913625000,       9185400000,      24494400000,       39191040000,       37623398400,       20065812480,        4586471424;
  6:    2562890625,     32805000000,     183708000000,     587865600000,     1175731200000,     1504935936000,     1203948748800,      550376570880,      110075314176;
  7:   38443359375,    553584375000,    3542940000000,   13226976000000,    31744742400000,    50791587840000,    54177693696000,    37150418534400,    14860167413760,     2641807540224;
  8:  576650390625,   9226406250000,   66430125000000,  283435200000000,   793618560000000,  1523747635200000,  2031663513600000,  1857520926720000,  1114512556032000,   396271131033600,    63403380965376;
  9: 8649755859375, 152235703125000, 1217885625000000, 5845851000000000, 18706723200000000, 41903059968000000, 67044895948800000, 76622738227200000, 61298190581760000, 32692368310272000, 10461557859287040, 1521681143169024;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (15+39*x+24*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 720*x + 576*x^2.
%t Mathematica
P[0, x_] := 225 + 720*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15 + 39*x + 24*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 225 + 720*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15 + 39*x + 24*x^2)*v[i - 1])/(1 + 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.