Real-rooted atlas

A137597 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + x\right)\,P_{n-1}(x) + \left(3 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2199129
310370151189
46225431692211357
541174351175533530110589
6295213640217896446059039188909
7227290318907183918561969765959491797085
81865881292533319281796944701487011927814706017583373
91623952328061620207446378573881217324834866335238215472176179037
10149142952281015445229694371106840780311898105945499746277606603045851804638925
111439618143293238640326207486911351857551445541745981322871468212914866179949541809918904557341
12145556317813182604230430823596508174302524886388175816159301065027617253433422796279341715680555991141102929260813
1315370065403635862742234937363989374422949347371692430818766256963772785056614706710865298711775354999852238953734513011239432543709
14169068488319141886771282454665908048257308926403494813543856830144147172789769119069559614584393824169999172214334420185875793538089464147115511264432973
Technical details
Node id
n-73540840b1c566dc · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-86218b05355409c1

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:       19,        9,        1;
  3:      103,       70,       15,       1;
  4:      622,      543,      169,      22,       1;
  5:     4117,     4351,     1755,     335,      30,      1;
  6:    29521,    36402,    17896,    4460,     590,     39,     1;
  7:   227290,   318907,   183918,   56196,    9765,    959,    49,    1;
  8:  1865881,  2925333,  1928179,  694470,  148701,  19278,  1470,   60,  1;
  9: 16239523, 28061620, 20744637, 8573881, 2173248, 348663, 35238, 2154, 72, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+x)*P(n-1,x) + (3+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[(4 + x)*P[n - 1, x] + (3 + 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] = (4 + x)*v[i - 1] + (3 + 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

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.