Real-rooted atlas

A013622 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-7 + 15\,x + 15\,n\right)\,P_{n}(x) = \left(64 + 240\,x + 225\,x^{2} + 120\,n + 225\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1231538
23046902251219
333921368015525337535972
434816203520410400310500506251009861
53399682611200763200010260000582187575937527424418
632112643059712011750400022896000023085000010479375011390625727306759
7296222723371827201606348800411264000060102000004847850000183389062517085937518948593792
82684354563554672640202309632006425395200012337920000014424480000096957000000314381250002562890625486890038921
923991418883623878656023994040320091039334400021685708800003331238400000324550800000018698850000005305183593753844335937512373135674398
102120640102435987128320027179089920001199702016000034139750400000650571264000008328096000000069546600000000350603437500008841972656250576650390625311599410433099
11185757335552349905616896029689380864000149484994560000494877081600000112661176320000017890709760000001963051200000000143439862500000064277296875000014589254882812586497558593757788184108166012
12161490770329633436320399360314915055206400178136285184000067268247552000001781557493760000033798352896000000460046822400000004416865200000000028687972500000000115699134375000002387332617187500129746337890625193420380856527181
13139500537774083149070021427203260041238937600204694785884160008684143902720000026234616545280000057900618547200000094152554496000000011213641296000000009569874600000000005594154637500000002051030109375000003879415502929687519461950683593754777388126180429978
141198467674275842929511293255680330652352249856002282028867256320001074647625891840000364734043914240000091821157908480000001737018556416000000024715045555200000000261651630240000000002009673666000000000010679749762500000000358930269140625000062667481201171875029192926025390625117440472331196900239
Technical details
Node id
n-af0f86ab2b07c466 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5ee05b1a0fa9fb0d

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:         23,          15;
  2:        304,         690,          225;
  3:       3392,       13680,        15525,         3375;
  4:      34816,      203520,       410400,       310500,         50625;
  5:     339968,     2611200,      7632000,     10260000,       5821875,        759375;
  6:    3211264,    30597120,    117504000,    228960000,     230850000,     104793750,      11390625;
  7:   29622272,   337182720,   1606348800,   4112640000,    6010200000,    4847850000,    1833890625,     170859375;
  8:  268435456,  3554672640,  20230963200,  64253952000,  123379200000,  144244800000,   96957000000,   31438125000,   2562890625;
  9: 2399141888, 36238786560, 239940403200, 910393344000, 2168570880000, 3331238400000, 3245508000000, 1869885000000, 530518359375, 38443359375;
%F Formula
The row polynomials satisfy (15*n-7+15*x)*P(n,x) = (120*n+64+(225*n+240)*x+225*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[((120*n + 64 + (225*n + 240)*x + 225*x^2)*P[n - 1, x])/(15*n - 7 + 15*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] = ((120*n + 64 + (225*n + 240)*x + 225*x^2)*v[i - 1])/(15*n - 7 + 15*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

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.