Real-rooted atlas

A013622 → basis A013611

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(8 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
182028
264320400784
351238409600800021952
4409640960153600256000160000614656
532768409600204800051200006400000320000017210368
62621443932160245760008192000015360000015360000064000000481890304
72097152367001602752512001146880000286720000043008000003584000000128000000013492928512
81677721633554432029360128001468006400045875200000917504000001146880000008192000000025600000000377801998336
9134217728301989888030198988800176160768000660602880000165150720000027525120000002949120000000184320000000051200000000010578455953408
1010737418242684354560030198988800020132659200008808038400000264241152000005505024000000078643200000000737280000000004096000000000010240000000000296196766695424
1185899345922362232012802952790016000221459251200001107296256000003875536896000009688842240000001730150400000000216268800000000018022400000000009011200000000002048000000000008293509467471872
1268719476736206158430208028346784153600236223201280000132875550720000053150220288000001550214758400000033218887680000000519045120000000005767168000000000043253760000000000196608000000000004096000000000000232218265089212416
1354975581388817867063951360268005959270400245672129331200015354508083200000690952863744000002303176212480000005757940531200000001079613849600000000149946368000000000014994636800000000001022361600000000000425984000000000000819200000000000006502111422497947648
14439804651110415393162788864025013889531904002501388953190400017197049053184000085985245265920000032244466974720000009212704849920000000201527918592000000003358798643200000000041984983040000000000381681664000000000002385510400000000000091750400000000000001638400000000000000182059119829942534144

Generated b-files: A009972 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-815b996b7a3cb30d · revision 1 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-9629d1e9c04dbfbe

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:         1;
  1:         8,         20;
  2:        64,        320,         400;
  3:       512,       3840,        9600,         8000;
  4:      4096,      40960,      153600,       256000,       160000;
  5:     32768,     409600,     2048000,      5120000,      6400000,       3200000;
  6:    262144,    3932160,    24576000,     81920000,    153600000,     153600000,      64000000;
  7:   2097152,   36700160,   275251200,   1146880000,   2867200000,    4300800000,    3584000000,    1280000000;
  8:  16777216,  335544320,  2936012800,  14680064000,  45875200000,   91750400000,  114688000000,   81920000000,   25600000000;
  9: 134217728, 3019898880, 30198988800, 176160768000, 660602880000, 1651507200000, 2752512000000, 2949120000000, 1843200000000, 512000000000;
%F Formula
The row polynomials satisfy P(n,x) = (8+20*x)*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[(8 + 20*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8 + 20*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, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-96f2019a633e1196
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-96f2019a633e1196
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-96f2019a633e1196

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

Same fingerprint (6)

Equal up to content, sign, reversal and a small index shift.

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  4. [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