Real-rooted atlas

A038259 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(6 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 7 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
07512
1426525132
22526004751251452
3151248605850312562515972
49072367205940048000193753125175692
554432265680540000585000356250115625156251932612
6326592186624045684006210000506250024750006718757812521258732
719595521283040036741600601020006142500040162500164062503828125390625233846052
811757312867801602846016005443200006690600005481000002992500001050000002148437519531252572306572
970543872579467520214151040046889280006735960000663390000045360000002126250000653906250119140625976562528295372292
10423263232382952448015746400000388411200006386040000073483200000603855000003543750000014554687500398437500065429687548828125311249095212
11253957939225093463040113626022400311778720000577368000000760201200000729729000000514552500000264515625000966796875002384765625035644531252441406253423740047332
1215237476352163258675200807223449600243880243200050231016000007448047200000817938000000067359600000004159856250000190265625000062648437500014062500000019287109375122070312537661140520652
139142485811210557394329605659634073600186689318400004233262176000069803791200000863165160000008131266000000058638937500000322152187500001327218750000039761718750008188476562501037597656256103515625414272545727172
1454854914867267915608883203923650160640014031176140800034734038976000063048585600000086691805200000091945854000000075839692500000048648600000000024070921875000090217968750000247939453125004716796875000555419921875305175781254556998002998892

Computed in your browser ?

Technical details
Node id
n-9b8d59f51ac4acc1 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-2a535ada15294c9b

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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

OEIS matches ?

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        7,         5;
  1:       42,        65,         25;
  2:      252,       600,        475,        125;
  3:     1512,      4860,       5850,       3125,        625;
  4:     9072,     36720,      59400,      48000,      19375,       3125;
  5:    54432,    265680,     540000,     585000,     356250,     115625,      15625;
  6:   326592,   1866240,    4568400,    6210000,    5062500,    2475000,     671875,      78125;
  7:  1959552,  12830400,   36741600,   60102000,   61425000,   40162500,   16406250,    3828125,    390625;
  8: 11757312,  86780160,  284601600,  544320000,  669060000,  548100000,  299250000,  105000000,  21484375,   1953125;
  9: 70543872, 579467520, 2141510400, 4688928000, 6735960000, 6633900000, 4536000000, 2126250000, 653906250, 119140625, 9765625;
%F Formula
The row polynomials satisfy P(n,x) = (6+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 5*x.
%t Mathematica
P[0, x_] := 7 + 5*x;
P[n_, x_] := P[n, x] = Expand[(6 + 5*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] = 7 + 5*x; for(n = 1, N, my(i = n + 1); v[i] = (6 + 5*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-90d99cba1db08a8a

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  2. [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
  3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.