Real-rooted atlas

A038319 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(224 + 360\,x + 100\,x^{2} + 80\,n + 100\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 896 + 1440\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
089614404002736
11945653760432008000124416
23932161556480215040011520001600005412096
37602176393216007782400071680000288000003200000228427776
4142606336912261120235929600031129600002150400000691200000640000009432723456
52617245696199648870406385827840011010048000010895360000060211200000161280000001280000000383113691136
647244640256418759311360159719096320034057748480004404019200000348651520000016056320000003686400000002560000000015359376162816
784181359001685040352460803768833802240095831457792000153259868160000158544691200000104595456000000412876800000008294400000000512000000000609359740010496

All 15 rows

Technical details
Node id
n-9688148ffd278232 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d37b0b195ff0bdee

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:             896,             1440,               400;
  1:           19456,            53760,             43200,              8000;
  2:          393216,          1556480,           2150400,           1152000,             160000;
  3:         7602176,         39321600,          77824000,          71680000,           28800000,            3200000;
  4:       142606336,        912261120,        2359296000,        3112960000,         2150400000,          691200000,           64000000;
  5:      2617245696,      19964887040,       63858278400,      110100480000,       108953600000,        60211200000,        16128000000,         1280000000;
  6:     47244640256,     418759311360,     1597190963200,     3405774848000,      4404019200000,      3486515200000,      1605632000000,       368640000000,       25600000000;
  7:    841813590016,    8504035246080,    37688338022400,    95831457792000,    153259868160000,    158544691200000,    104595456000000,     41287680000000,     8294400000000,      512000000000;
  8:  14843406974976,  168362718003200,   850403524608000,  2512555868160000,   4791572889600000,   6130394726400000,   5284823040000000,   2988441600000000,  1032192000000000,   184320000000000,   10240000000000;
  9: 259484744155136, 3265549534494720, 18519898980352000, 62362925137920000, 138190572748800000, 210829207142400000, 224781139968000000, 166094438400000000, 82182144000000000, 25231360000000000, 4055040000000000, 204800000000000;
%F Formula
The row polynomials satisfy (5*n+9+5*x)*P(n,x) = (80*n+224+(100*n+360)*x+100*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 896 + 1440*x + 400*x^2.
%t Mathematica
P[0, x_] := 896 + 1440*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((80*n + 224 + (100*n + 360)*x + 100*x^2)*P[n - 1, x])/(5*n + 9 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 896 + 1440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 224 + (100*n + 360)*x + 100*x^2)*v[i - 1])/(5*n + 9 + 5*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.