Real-rooted atlas

A130123 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1101626
2100320256676
3100048007680409617576
4100006400015360016384065536456976
5100000800000256000040960003276800104857611881376
6100000096000003840000081920000983040006291456016777216308915776
71000000011200000053760000014336000002293760000220200960011744051202684354568031810176
81000000001280000000716800000022937600000458752000005872025600046976204800214748364804294967296208827064576
91000000000144000000009216000000034406400000082575360000013212057600001409286144000966367641600386547056640687194767365429503678976
101000000000016000000000011520000000004915200000000137625600000002642411520000035232153600000322122547200001932735283200068719476736001099511627776141167095653376
1110000000000017600000000001408000000000067584000000000216268800000000484442112000000775107379200000885837004800000708669603840000377957122048000120946279055360175921860444163670344486987776
1210000000000001920000000000016896000000000090112000000000032440320000000008304721920000000155021475840000002126008811520000021260088115200000151182848819200007256776743321600211106232532992028147497671065695428956661682176
1310000000000000208000000000000199680000000000011714560000000000468582400000000001349517312000000002878970265600000004606352424960000005527622909952000004913442586624000003144603255439360001372190511464448003659174697238528045035996273704962481152873203736576
14100000000000000224000000000000023296000000000000149094400000000000656015360000000000209924915200000000050381979648000000009212704849920000000128977867898880000001375763924254720000011006111394037760000640355572016742400025614222880669696006305039478318694407205759403792793664509974703297150976
Technical details
Node id
n-b7c94357354b5f47 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         10,          16;
  2:        100,         320,         256;
  3:       1000,        4800,        7680,         4096;
  4:      10000,       64000,      153600,       163840,        65536;
  5:     100000,      800000,     2560000,      4096000,      3276800,       1048576;
  6:    1000000,     9600000,    38400000,     81920000,     98304000,      62914560,      16777216;
  7:   10000000,   112000000,   537600000,   1433600000,   2293760000,    2202009600,    1174405120,    268435456;
  8:  100000000,  1280000000,  7168000000,  22937600000,  45875200000,   58720256000,   46976204800,  21474836480,   4294967296;
  9: 1000000000, 14400000000, 92160000000, 344064000000, 825753600000, 1321205760000, 1409286144000, 966367641600, 386547056640, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (10+16*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[(10 + 16*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] = (10 + 16*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. [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. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [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
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.