Real-rooted atlas

A136124 → 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

\(P_{n}(x) = \left(4 + 2\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
23022456
3210214728504
4168021321004208165040
515120225481330038805603255440
61512002557201780966540013360144064665280
716632003115320247049610755922777604256035841288648640
819958400407102403587659217848096548430410662401281288704256121080960
9259459200569149920547816176303778432106992144248297283798144369408207365121816214400
103632428800848701728088077263045348530400210544688056160048010283347212768000102912048640102429059430400
115448643200013457011680014908992912097843408608422787640001263490096026657030403971869444097280027878401126402048494010316800
1287178291200022620947328002654579099520186367439596887214704121628671594336067921050560116863971841449938688126551040737792025804840968892185702400
13148203095040004019917628160049652034157440369916229304961855384849260866184651195521728089746240334510853248480217520645051245056378526720191426565857288192168951528345600
142667655710720007532257920768009741349673971207651532810638084079525187279361562400691371524434254567142494773748509441533413243648186965915136169159710721101621248488284161318912163843379030566912000
Technical details
Node id
n-d008e423f598159c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f01da5544602b4b6

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:         5,         2;
  2:        30,        22,         4;
  3:       210,       214,        72,         8;
  4:      1680,      2132,      1004,       208,        16;
  5:     15120,     22548,     13300,      3880,       560,       32;
  6:    151200,    255720,    178096,     65400,     13360,     1440,      64;
  7:   1663200,   3115320,   2470496,   1075592,    277760,    42560,    3584,    128;
  8:  19958400,  40710240,  35876592,  17848096,   5484304,  1066240,  128128,   8704,   256;
  9: 259459200, 569149920, 547816176, 303778432, 106992144, 24829728, 3798144, 369408, 20736, 512;
%F Formula
The row polynomials satisfy P(n,x) = (n+4+2*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 + 4 + 2*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 + 4 + 2*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.