Real-rooted atlas

A272866 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 10\,x + 5\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1410519
2168014010025361
3644801440220018007501256859
4256256010880256003640032000170005000625130321
5102412800704002240004560006200005700003500001375003125031252476099
640966144041472016640004416000816000010760000102000006900000325000010125001875001562547045881
716384286720229376011110400363776008512000014672000018920000018340000013300000071050000271250007000000109375078125893871739

All 15 rows

Technical details
Node id
n-6494b967cfaa76e5 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c64e29f1cee5cd50

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,      10,        5;
  2:     16,      80,      140,       100,         25;
  3:     64,     480,     1440,      2200,       1800,        750,         125;
  4:    256,    2560,    10880,     25600,      36400,      32000,       17000,        5000,         625;
  5:   1024,   12800,    70400,    224000,     456000,     620000,      570000,      350000,      137500,       31250,        3125;
  6:   4096,   61440,   414720,   1664000,    4416000,    8160000,    10760000,    10200000,     6900000,     3250000,     1012500,      187500,       15625;
  7:  16384,  286720,  2293760,  11110400,   36377600,   85120000,   146720000,   189200000,   183400000,   133000000,    71050000,    27125000,     7000000,     1093750,      78125;
  8:  65536, 1310720, 12124160,  68812800,  268083200,  759808000,  1619968000,  2649600000,  3359200000,  3312000000,  2531200000,  1484000000,   654500000,   210000000,   46250000,    6250000,    390625;
  9: 262144, 5898240, 61931520, 403046400, 1821081600, 6064128000, 15418368000, 30597120000, 48032640000, 60088000000, 60040800000, 47808000000, 30114000000, 14805000000, 5557500000, 1537500000, 295312500, 35156250, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (4+10*x+5*x^2)*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 + 10*x + 5*x^2)*P[n - 1, 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 + 10*x + 5*x^2)*v[i - 1]); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.