Real-rooted atlas

A038324 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(31 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1311041
29616201001681
329791288309300100068921
49235211191640576600124000100002825761
528629151461760502979100096100001550000100000115856201
6887503681171774906013852815005958200001441500001860000010000004750104241
72751261411162125257670601212171003232323500010426850000201810000021700000010000000194754273881
88528910374412201009128880248501030680016032324560006464647000001668296000002690800000024800000001000000007984925229121
92643962216067176760193369690990454107996007455030920400036072730260000116363646000002502444000000345960000000279000000001000000000327381934393961
1081962828698080126439622160671003838009668484500330151369332000018637577301000007214546052000001939394100000003574920000000043245000000003100000000001000000000013422659310152401
1125408476896404831901591115678881101454179218836905001407270211777650009079162656630000041002670062200000132266677620000003047619300000000491551500000000528550000000003410000000000100000000000550329031716248441
127876627837885497613049017227568579720540954669407328660058167168753476200004221810635332950000217899903759120000082005340124400000022674287592000000045714289500000000655402000000000063426000000000037200000000000100000000000022563490300366186081
13244175462974450425911023961618925114689301981861197919576818002344136900765090860001890432984487976500001097670765186567000004721164581447600000015229563165960000000368457173370000000066031751500000000085202260000000000749580000000000040300000000000010000000000000925103102315013629321
14756943935220796320321341845648164230596274071677313324758028251009248685590291358484000820447915267781801000052932123565663342000002561231785435323000000944232916289520000000266517355404300000000573155603020000000009244445210000000000108439240000000000087451000000000000434000000000000010000000000000037929227194915558802161
Technical details
Node id
n-69cb08626b22511e · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a4bc2de2f0f9f406

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:             31,             10;
  2:            961,            620,            100;
  3:          29791,          28830,           9300,           1000;
  4:         923521,        1191640,         576600,         124000,          10000;
  5:       28629151,       46176050,       29791000,        9610000,        1550000,         100000;
  6:      887503681,     1717749060,     1385281500,      595820000,      144150000,       18600000,       1000000;
  7:    27512614111,    62125257670,    60121217100,    32323235000,    10426850000,     2018100000,     217000000,     10000000;
  8:   852891037441,  2201009128880,  2485010306800,  1603232456000,   646464700000,   166829600000,   26908000000,   2480000000,   100000000;
  9: 26439622160671, 76760193369690, 99045410799600, 74550309204000, 36072730260000, 11636364600000, 2502444000000, 345960000000, 27900000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (31+10*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[(31 + 10*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] = (31 + 10*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

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.