Real-rooted atlas

A136124 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2209130
312074151210
48406381792211680
567205944207035530115120
66048060216245745265625391151200
7604800662640305956772241151510154911663200
8665280078938404028156115542020388922680155460119958400
97983360010137888056231712178931963602088476049413282274721259459200
101037836800139775904083239113628884326064720340979072510133137089032108513632428800
1114529715200206064633601305123494448761967761194928020201790490239771072005773115830440099154486432000
122179457280003236266656002163749875208619418658422800117076422178537056144709554063702374322318183058851141871782912000
1334871316480005395972377600378562646592015954819728644509960598009034868299613204938890142646632711395527066525032759907709130114820309504000
1459281238016000952186620672006975162229824030908820004608926241498946419869236707323148326441263745486644933637059172270478211134433340704399191471266765571072000
Technical details
Node id
n-ee1493164ea145b8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6546c73f6027554e

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:        4,         1;
  2:       20,         9,        1;
  3:      120,        74,       15,        1;
  4:      840,       638,      179,       22,       1;
  5:     6720,      5944,     2070,      355,      30,      1;
  6:    60480,     60216,    24574,     5265,     625,     39,     1;
  7:   604800,    662640,   305956,    77224,   11515,   1015,    49,    1;
  8:  6652800,   7893840,  4028156,  1155420,  203889,  22680,  1554,   60,  1;
  9: 79833600, 101378880, 56231712, 17893196, 3602088, 476049, 41328, 2274, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (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[(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] = (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.