Real-rooted atlas

A136124 → basis A008277 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
22110132
313693181248
410459162582812248
592769805358057040123272
69328911448650895105801095541270400
7104737614544257569661928152618019117013479744
8129755612001440811873428356109659507057176310888149079936
91757211402969097211969256886767014813368096158571011373647881081752299136
1025812845414726308610345380548513331204403028563304259329237802102103607065130112443966464
1140864292184803835246616398655769027328278285699750084011348101381196700128254290366960100651541220823265280
1269334790742114549738021401250139840246583885570180165954844935304117236723705660420304200072167849556100601392618014182740958208
13125485403208762792818222429325709147637108130077850534064055041087240826835004567114065653416108365417167125295323217571597411618798208184201179220992
142412533676791855666741156323665554253424895633020850130541881022978229082812292980510821035262008251473798428908562893936145115588392825868362515029562484323811794602069000192
Technical details
Node id
n-5bc9266c450b7176 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-58fdbf6ede9b43e4

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         4,         1;
  2:        21,        10,         1;
  3:       136,        93,        18,        1;
  4:      1045,       916,       258,       28,        1;
  5:      9276,      9805,      3580,      570,       40,       1;
  6:     93289,    114486,     50895,    10580,     1095,      54,      1;
  7:   1047376,   1454425,    756966,   192815,    26180,    1911,     70,    1;
  8:  12975561,  20014408,  11873428,  3561096,   595070,   57176,   3108,   88,   1;
  9: 175721140, 296909721, 196925688, 67670148, 13368096, 1585710, 113736, 4788, 108, 1;
%F Formula
The row polynomials satisfy P(n,x) = (n+3+x)*P(n-1,x) + (1+x)*d/dx 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] + (1 + x)*D[P[n - 1, x], 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] + (1 + x)*deriv(v[i - 1], 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.