Real-rooted atlas

A038221 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(6 + 21\,x + 15\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161521
236180225441
32161620405033759261
4129612960486008100050625194481
5777697200486000121500015187507593754084101
64665669984043740001458000027337500273375001139062585766121
72799364898880367416001530900003827250005740875004784062501708593751801088541
8167961633592320293932800146966400045927000009185400000114817500008201250000256289062537822859361
9100776962267481602267481600132269760004960116000012400290000020667150000022143375000013839609375038443359375794280046581
10604661761511654400170061120001133740800004960116000001488034800000310007250000044286750000004151882812500230660156250057665039062516679880978201
11362797056997691904012471148800093533616000046766808000001636838280000040920957000000730731375000009134142187500076117851562500380589257812508649755859375350277500542221
1221767823366530347008089792271360074826892800004209012720000016836050880000049105148400000010522531800000001644145593750000182682843750000013701213281250006227824218750001297463378906257355827511386641
1313060694016424472555520636708833280058364976384000364781102400000164151496080000054717165360000001367929134000000025648671262500000356231545312500003562315453125000024288514453125000101202143554687501946195068359375154472377739119461
147836416409627427457433604456961832960044569618329600030641612601600001532080630080000057453023628000000164151496080000000359081397675000000598468996125000000748086245156250000680078404687500000425049002929687500163480385742187500291929260253906253243919932521508681
Technical details
Node id
n-d5d11fe45846957a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,        15;
  2:       36,       180,        225;
  3:      216,      1620,       4050,        3375;
  4:     1296,     12960,      48600,       81000,       50625;
  5:     7776,     97200,     486000,     1215000,     1518750,       759375;
  6:    46656,    699840,    4374000,    14580000,    27337500,     27337500,     11390625;
  7:   279936,   4898880,   36741600,   153090000,   382725000,    574087500,    478406250,    170859375;
  8:  1679616,  33592320,  293932800,  1469664000,  4592700000,   9185400000,  11481750000,   8201250000,   2562890625;
  9: 10077696, 226748160, 2267481600, 13226976000, 49601160000, 124002900000, 206671500000, 221433750000, 138396093750, 38443359375;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (6+21*x+15*x^2)*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[Together[((6 + 21*x + 15*x^2)*P[n - 1, x])/(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] = ((6 + 21*x + 15*x^2)*v[i - 1])/(1 + x)); 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.