Real-rooted atlas

A038259 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(121 + 110\,x + 25\,x^{2} + 55\,n + 25\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116521
223116025416
33146346512001257936
44126162920346508000625147456
552707610315257865002887505000031252686976
666030911581228015472875786500021656253000001562548234496
7814918062311081852767149001805168756881875015159375175000078125855638016
899384572132596722404622163700368953200018051687505505500001010625001000000039062515032385536
91200409733644723057445733426254006933245550041507235000162465187504129125000649687500562500001953125261993005056
101438348091516002048668001118076436125122237709000086665569375041507235000013538765625029493750000406054687531250000097656254535485464576
11171187002366679109145033051650563383700020498067995625168076849875009533212631250380482987500010637601562502027695312502481445312517187500004882812578065325572096
122025712861338110271220141996023732743509915033011267674000030747101993437520169221985000095332126312500326128275000007978201171875135179687500014888671875093750000002441406251337006139375616
1323852055663079613167133598697653338146546148700514209442714825053643309970250003997123259146875218499904837500088522688718750026497922343750057620341796875878667968750087978515625050781250000122070312522799473113563136
14279633968365841116696438964155720460849675954417757789008607680300089986652475094375751006339583500004663310469004687521849990483750000774573526289062520609495156250004033423925781255591523437500051320800781252734375000006103515625387309567953862656
Technical details
Node id
n-d67b30ee666d0497 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-42dc7898d6edde45

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:          16,           5;
  2:         231,         160,          25;
  3:        3146,        3465,        1200,         125;
  4:       41261,       62920,       34650,        8000,         625;
  5:      527076,     1031525,      786500,      288750,       50000,        3125;
  6:     6603091,    15812280,    15472875,     7865000,     2165625,      300000,      15625;
  7:    81491806,   231108185,   276714900,   180516875,    68818750,    15159375,    1750000,     78125;
  8:   993845721,  3259672240,  4622163700,  3689532000,  1805168750,   550550000,  101062500,  10000000,   390625;
  9: 12004097336, 44723057445, 73342625400, 69332455500, 41507235000, 16246518750, 4129125000, 649687500, 56250000, 1953125;
%F Formula
The row polynomials satisfy (5*n+6+5*x)*P(n,x) = (55*n+121+(25*n+110)*x+25*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[((55*n + 121 + (25*n + 110)*x + 25*x^2)*P[n - 1, x])/(5*n + 6 + 5*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] = ((55*n + 121 + (25*n + 110)*x + 25*x^2)*v[i - 1])/(5*n + 6 + 5*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.