Real-rooted atlas

A038336 → x ↦ x+1 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 30\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 1920\,x + 900\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0102419209003844
132768921608640027000238328
2104857639321605529600345600081000014776336
33355443215728640029491200027648000012960000024300000916132832
410737418246039797760141557760001769472000012441600000466560000072900000056800235584
534359738368225485783040634178764800990904320000928972800000522547200000163296000000218700000003521614606208
61099511627776824633720832027058293964800507343011840005945425920000044590694400000209018880000005598720000000656100000000218340105584896
735184372088832296868139499520111325552312320024352464568320003424565329920000321052999680000020065812480000008062156800000001889568000000001968300000000013537086546263552

All 15 rows

Technical details
Node id
n-2e9b5735d80321d0 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d935ec63feee39a0

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1024,               1920,                 900;
  1:             32768,              92160,               86400,               27000;
  2:           1048576,            3932160,             5529600,             3456000,              810000;
  3:          33554432,          157286400,           294912000,           276480000,           129600000,             24300000;
  4:        1073741824,         6039797760,         14155776000,         17694720000,         12441600000,           4665600000,            729000000;
  5:       34359738368,       225485783040,        634178764800,        990904320000,        928972800000,         522547200000,         163296000000,         21870000000;
  6:     1099511627776,      8246337208320,      27058293964800,      50734301184000,      59454259200000,       44590694400000,       20901888000000,       5598720000000,        656100000000;
  7:    35184372088832,    296868139499520,    1113255523123200,    2435246456832000,    3424565329920000,     3210529996800000,     2006581248000000,     806215680000000,     188956800000000,      19683000000000;
  8:  1125899906842624,  10555311626649600,   44530220924928000,  111325552312320000,  182643484262400000,   205473919795200000,   160526499840000000,   85996339200000000,   30233088000000000,    6298560000000000,    590490000000000;
  9: 36028797018963968, 371546969258065920, 1741626418397184000, 4898324301742080000, 9184358065766400000, 12054469961318400000, 11301065588736000000, 7567677849600000000, 3547348992000000000, 1108546560000000000, 207852480000000000, 17714700000000000;
%F Formula
The row polynomials satisfy P(n,x) = (32+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 1920*x + 900*x^2.
%t Mathematica
P[0, x_] := 1024 + 1920*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 30*x)*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] = 1024 + 1920*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 30*x)*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.