Real-rooted atlas

A013609 → basis A013609 → basis A008277

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(7 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
2497216137
3343988528641923
424011224011872332825630097
516807144244227040110720192001024519035
611764916539123982576301184088448010444840969759001
782354318663628663542887315526432390400635801654476816384198306291
85764801208594080106996883216553671681037600256301228032423198722752512655364323661089
940353607231759408416883908800357318515203048627456012283557888251682816026561740813565952262144100539814123
10282475249256549493522625290083367464408140808441117209604536011612161271559045121936392192015915417606553600010485762480798081961
1119773267432833343438684040554922448152324915008642240033283840015623678236672575623768473611863608852481395258163209184215040311427072419430464673993391715
12138412872013124587089520617416612121923055795582062085761376212866565114406437437442409380811735046454758172262410186950377472953024512000513739325441459617792167772161774716400938673
139688901040734425823133524938623266540960605297825076672014473483522417920161134478662277129514843091353600322291768498585665548139161190481727855525888622667287756827993623756867612180486710886451094539079016539
1467822307284937907161051039214217052290648496118761079830731520357102034018679040492957026477819904359413927296872448150861491339919360384554450128076806136223354060800619565047349248391834396262401491611942912310042951682684354561538944300826771513
Technical details
Node id
n-8270042dd6030930 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-01a7a97b7ee8f4b3

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:        7,          4;
  2:       49,         72,          16;
  3:      343,        988,         528,          64;
  4:     2401,      12240,       11872,        3328,         256;
  5:    16807,     144244,      227040,      110720,       19200,        1024;
  6:   117649,    1653912,     3982576,     3011840,      884480,      104448,       4096;
  7:   823543,   18663628,    66354288,    73155264,    32390400,     6358016,     544768,     16384;
  8:  5764801,  208594080,  1069968832,  1655367168,  1037600256,   301228032,   42319872,   2752512,    65536;
  9: 40353607, 2317594084, 16883908800, 35731851520, 30486274560, 12283557888, 2516828160, 265617408, 13565952, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) + 4*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[(7 + 4*x)*P[n - 1, x] + 4*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] = (7 + 4*x)*v[i - 1] + 4*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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

Proof chains ? JSON

Same fingerprint ?

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.