Real-rooted atlas

A156992 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(5 + 3\,x + 5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110616
215018054384
330005400324064812288
475000180000162000648009720491520
52250000675000081000004860000145800017496023592960
6787500002835000004252500003402000001530900003674160036741601321205760
7315000000013230000000238140000002381400000014288400000514382400010287648008817984084557168640
81417500000006804000000001428840000000171460800000012859560000006172588800001851776640003174474240023808556806088116142080
97087500000000382725000000009185400000000012859560000000011573604000000069441624000000277766496000007142567040000107138505600071425670400487049291366400
1038981250000000023388750000000006314962500000000101039400000000001060913700000000076385786400000003819289320000000130947062400000029463089040000039284118720000235704712320042860337640243200
1123388750000000000154365750000000000463097250000000000833575050000000000100029006000000000084024365040000000050414619024000000021606265296000000064818795888000000129637591776000001555651101312000848536964352004114592413463347200
12152026875000000000010945935000000000000361215855000000000007224317100000000000097528280850000000000936271496160000000006553900473120000000033705773861760000000126396651981600000003370577386176000000606703929511680000661858832194560003309294160972800427917611000188108800
131064188125000000000008300667375000000000002988240255000000000000657412856100000000000098611928415000000000001065008826882000000000085200706150560000000005112042369033600000000230041906606512000000076680635535504000000018403352528520960000030114576864852480000301145768648524800013899035476085760047926772432021068185600
147981410937500000000000670438518750000000000002614710223125000000000006275304535500000000000001035425248357500000000000124251029802900000000000011182592682261000000000007668063553550400000000004025733365613960000000001610293346245584000000004830880038736752000000010540101902698368000000158101528540475520000014593987249890048000062545659642385920005751212691842528182272000
Technical details
Node id
n-cb1e68669081ca55 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            10,              6;
  2:           150,            180,             54;
  3:          3000,           5400,           3240,             648;
  4:         75000,         180000,         162000,           64800,            9720;
  5:       2250000,        6750000,        8100000,         4860000,         1458000,         174960;
  6:      78750000,      283500000,      425250000,       340200000,       153090000,       36741600,        3674160;
  7:    3150000000,    13230000000,    23814000000,     23814000000,     14288400000,     5143824000,     1028764800,      88179840;
  8:  141750000000,   680400000000,  1428840000000,   1714608000000,   1285956000000,   617258880000,   185177664000,   31744742400,    2380855680;
  9: 7087500000000, 38272500000000, 91854000000000, 128595600000000, 115736040000000, 69441624000000, 27776649600000, 7142567040000, 1071385056000, 71425670400;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+5+(3*n+3)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5*n + 5 + (3*n + 3)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5*n + 5 + (3*n + 3)*x)*v[i - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.