Real-rooted atlas

A038336 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(32 + 10\,x\right)\,P_{n-1}(x) + \left(10\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1024 + 740\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
010247401001864
1327684132012600100087688
2104857620631201068400188000100004378096
3335544329713680076188000223400002600000100000231919232
4107374182444152899204933144000214696000041060000034200000100000013014935744
53435973836819617959488030067638720018244296000051032800000691040000043400000010000000772045880448
610995116277768583140368640175969680832001431822739200054987912000001076980800000109032000000536000000010000000048288111471616
735184372088832371487011760640100087374401280010636997791360005390952403200001433003376000002080075200000016370400000006480000000010000000003176144076918272

All 15 rows

Technical details
Node id
n-2ac15cdf83efabfa · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b233cd7cdc365023

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:              1024,                740,                 100;
  1:             32768,              41320,               12600,                1000;
  2:           1048576,            2063120,             1068400,              188000,               10000;
  3:          33554432,           97136800,            76188000,            22340000,             2600000,              100000;
  4:        1073741824,         4415289920,          4933144000,          2146960000,           410600000,            34200000,            1000000;
  5:       34359738368,       196179594880,        300676387200,        182442960000,         51032800000,          6910400000,          434000000,          10000000;
  6:     1099511627776,      8583140368640,      17596968083200,      14318227392000,       5498791200000,       1076980800000,       109032000000,        5360000000,        100000000;
  7:    35184372088832,    371487011760640,    1000873744012800,    1063699779136000,     539095240320000,     143300337600000,     20800752000000,     1637040000000,      64800000000,      1000000000;
  8:  1125899906842624,  15954298214835200,   55760304806272000,   75958123746560000,   49451855094400000,   17141580086400000,   3346672560000000,   374985600000000,   23628000000000,    770000000000,   10000000000;
  9: 36028797018963968, 681339524091504640, 3059078832074496000, 5267006720349440000, 4320114804262400000, 1900128118028800000, 479309676384000000, 71715256800000000, 6396192000000000, 330220000000000, 9020000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (32+10*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1024 + 740*x + 100*x^2.
%t Mathematica
P[0, x_] := 1024 + 740*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 10*x)*P[n - 1, x] + 10*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] = 1024 + 740*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 10*x)*v[i - 1] + 10*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.