Real-rooted atlas

A013612 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(21 + 15\,x\right)\,P_{n-1}(x) + \left(15\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 441 + 855\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04418552251521
192613739524300337574331
219448114851351800225587250506254117716
34084101563820751140885006576187512909375759375253985301
48576612120910162156664244625605161125020320875002665406251139062517202656961
51801088541765630755553712417191004993700118752553732562505606921250052624687501708593751265851691946
6378228593612783287048095200817738074253852704657025028175783934375921324324375014251722187501004653125002562890625100347157885131
77942800465811007656766218351065919769900100284401168074787528601441972381251307108110415625296392764937500340362126562501868347265625384433593758511079483188891

All 15 rows

Technical details
Node id
n-e75907fb73d769ed · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b4a8592e069a96fb

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             441,                855,                 225;
  1:            9261,              37395,               24300,                 3375;
  2:          194481,            1485135,             1800225,               587250,                50625;
  3:         4084101,           56382075,           114088500,             65761875,             12909375,               759375;
  4:        85766121,         2091016215,          6664244625,           6051611250,           2032087500,            266540625,            11390625;
  5:      1801088541,        76563075555,        371241719100,         499370011875,         255373256250,          56069212500,          5262468750,           170859375;
  6:     37822859361,      2783287048095,      20081773807425,       38527046570250,       28175783934375,        9213243243750,       1425172218750,        100465312500,         2562890625;
  7:    794280046581,    100765676621835,    1065919769900100,     2844011680747875,     2860144197238125,     1307108110415625,     296392764937500,      34036212656250,      1868347265625,       38443359375;
  8:  16679880978201,   3639478559084775,   55873393414232625,   203693567477861250,   274331855187506250,   168384541558471875,   52506218564296875,    8734454268750000,    773980154296875,    34022373046875,    576650390625;
  9: 350277500542221, 131271426341724915, 2904135242512135500, 14281876354752331875, 25276283782355925000, 20279893817425893750, 8353958384014031250, 1888134516326953125, 240148015787109375, 16917192509765625, 608942812500000, 8649755859375;
%F Formula
The row polynomials satisfy P(n,x) = (21+15*x)*P(n-1,x) + 15*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 441 + 855*x + 225*x^2.
%t Mathematica
P[0, x_] := 441 + 855*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 15*x)*P[n - 1, x] + 15*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] = 441 + 855*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 15*x)*v[i - 1] + 15*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. [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.