Real-rooted atlas

A147716 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141519
216120225361
364720270033756859
42563840216005400050625130321
510241920014400054000010125007593752476099
6409692160864000432000012150000182250001139062547045881
716384430080483840030240000113400000255150000318937500170859375893871739
865536196608025804800193536000907200000272160000051030000005467500000256289062516983563041
92621448847360132710400116121600065318400002449440000061236000000984150000009226406250038443359375322687697779
101048576393216006635520006635520000435456000001959552000006123600000001312200000000184528125000015377343750005766503906256131066257801
11419430417301504032440320003649536000027371520000014370048000005388768000000144342000000002706412500000033830156250000253726171875008649755859375116490258898219
12167772167549747201557135360019464192000016422912000009853747200000431101440000001385683200000003247695000000005412825000000006089428125000004151882812500001297463378906252213314919066161
1367108864327155712073610035200101213798400094887936000006404935680000032024678400000012009254400000003377602800000000703667250000000010555008750000000107948953125000006746809570312500194619506835937542052983462257059
142684354561409286144034351349760051527024640005313724416000039852933120000022417274880000009607403520000000315242928000000007881073200000000014777012250000000020150471250000000018891066796875000010898692382812500029192926025390625799006685782884121
Technical details
Node id
n-440a893d3788ec29 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-1bc3b9efcd292160

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:      4,      15;
  2:     16,     120,       225;
  3:     64,     720,      2700,       3375;
  4:    256,    3840,     21600,      54000,      50625;
  5:   1024,   19200,    144000,     540000,    1012500,      759375;
  6:   4096,   92160,    864000,    4320000,   12150000,    18225000,    11390625;
  7:  16384,  430080,   4838400,   30240000,  113400000,   255150000,   318937500,   170859375;
  8:  65536, 1966080,  25804800,  193536000,  907200000,  2721600000,  5103000000,  5467500000,  2562890625;
  9: 262144, 8847360, 132710400, 1161216000, 6531840000, 24494400000, 61236000000, 98415000000, 92264062500, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (4+15*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[(4 + 15*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] = (4 + 15*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.