Real-rooted atlas

A136572 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21824850
316232421648750
4194451845184230438415000
5291609720012960086400288003840375000
652488020995203499200311040015552004147204608011250000
7110224805143824010287648011430720076204800304819206773760645120393750000
8264539520141087744032920473604389396480365783040019508428806502809601238630401032192015750000000
971425670404285540224011428107264017777055744017777055744011851370496052672757760150493593602508226560185794560708750000000
10214277011200142851340800042855402240007618738176000888852787200071108222976003950456832000150493593600037623398400055738368000371589120035437500000000
1170711413696005185503671040017285012236800034570024473600046093365964800043020474900480028680316600320013657293619200045524312064000101165137920001348868505600817496064001949062500000000
1225456108930560020364887144448007467125286297600165936117473280002489041762099200026549778795724800206498279522304001179990168698880049166257029120001456777986048000291355597209600353158299648001961990553600116943750000000000
13992788248291840086041648185292800344166592741171200841296115589529600140216019264921600016825922311790592001495637538825830400997091692550553600498545846275276800184646609731584000492390959284224008952562896076800994729210675200510117543936007601343750000000000
14416971064282572800389172993330401280016864163044317388800449711014515130368008244701932777390080010992935910369853440010992935910369853440083755702174246502400488574929349771264002171444130443427840072381471014780928001754702327631052800292450387938508800299949115834368001428329123020800532094062500000000000
Technical details
Node id
n-1916a93c76107e9a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f06e51c139cfae05

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:          3,           2;
  2:         18,          24,            8;
  3:        162,         324,          216,           48;
  4:       1944,        5184,         5184,         2304,          384;
  5:      29160,       97200,       129600,        86400,        28800,         3840;
  6:     524880,     2099520,      3499200,      3110400,      1555200,       414720,       46080;
  7:   11022480,    51438240,    102876480,    114307200,     76204800,     30481920,     6773760,      645120;
  8:  264539520,  1410877440,   3292047360,   4389396480,   3657830400,   1950842880,   650280960,   123863040,   10321920;
  9: 7142567040, 42855402240, 114281072640, 177770557440, 177770557440, 118513704960, 52672757760, 15049359360, 2508226560, 185794560;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+2*n*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[(3*n + 2*n*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] = (3*n + 2*n*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.