Real-rooted atlas

A193685 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6 + x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1617
23713151
3235130211387
4154011873013013059
51042710451368557540125179
67301390954416569020980511215675
75290327930034512621267561907515476311920739
8396719569825774783367166836632524136470231076117765603
9307857476241196850254611210996255103126740271647223306901170463467
1024712645056817874552774773126056374075832240136344391540077108420457510511694736523
11205093744552798731275571851813317615035110870583352350515313287430329861371734156160121117436962739
121758549779750153752960593831386843850543831215221991356386987099465044612673081404546938726702581071381185448962291
131556667397424874280458856407351773684668200489162100747068426169324904612186794334164425947415205604795558323982551046515612036411042363
14142142848433748491334156737013769611577568241444498028760333625669618212123522194445973723477861587238590466202993760771604302757803213286175123062998801947
Technical details
Node id
n-9a5366d56b9a68b8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-790fafd8a27641c8

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:        6,        1;
  2:       37,       13,        1;
  3:      235,      130,       21,        1;
  4:     1540,     1187,      301,       30,       1;
  5:    10427,    10451,     3685,      575,      40,      1;
  6:    73013,    90954,    41656,     9020,     980,     51,     1;
  7:   529032,   793003,   451262,   126756,   19075,   1547,    63,    1;
  8:  3967195,  6982577,  4783367,  1668366,  325241,  36470,  2310,   76,  1;
  9: 30785747, 62411968, 50254611, 21099625, 5103126, 740271, 64722, 3306, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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

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.