Real-rooted atlas

A136572 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x + 6\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(n + 6\,n x + 12\,n x^{2} + 8\,n x^{3} + 6\,n^{2} x + 20\,n^{2} x^{2} + 16\,n^{2} x^{3} + 4\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
223272106
361446487681566
424768518412288960027864
5120480043200153600240000138240579960
672034560388800184320043200004976640225792013821840
750402822403810240225792007056000012192768011063808041287680371090160
840320258048040642560289013760112896000026011238403540418560264241152083607552011081266560
93628802612736047029248039016857601828915200052672757760955913011201070176665606772211712018579456000364270919040
1036288002903040005878656000557383680003048192000001053455155200238978252800035672555520003386105856000185794560000044962283520013070897683200
1139916800351267840079035264000843042816000526901760000021244678963200578327371776001079094804480001365729361920001124057088000005440436305920011771943321600508336496236800
12479001600459841536001138107801600134886850560009484231680000043703339581440013879856922624003107793036902400491662570291200053954740224000003917114140262400169515983831040033157640355840021298277805235200
13622702080064761016320017485474406400227958777446400178092794880000092323304865792003350994028462080087536170539417600166181948758425600227958777446400000220664096568115200143241006337228800560364122013696009998303861145600956386012521139200
148717829120097639686144002855960819712004061810943590400349061877964800002010596417077248008209935369732096002451012775103692800542861032610856960089359840758988800001081254073183764480093584124140322816005491568395734220800195966755678453760032137405267968000045820486952829388800
Technical details
Node id
n-5641edb36703d6ea · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      1,        8;
  2:      2,       32,        72;
  3:      6,      144,       648,        768;
  4:     24,      768,      5184,      12288,        9600;
  5:    120,     4800,     43200,     153600,      240000,      138240;
  6:    720,    34560,    388800,    1843200,     4320000,     4976640,     2257920;
  7:   5040,   282240,   3810240,   22579200,    70560000,   121927680,   110638080,     41287680;
  8:  40320,  2580480,  40642560,  289013760,  1128960000,  2601123840,  3540418560,   2642411520,   836075520;
  9: 362880, 26127360, 470292480, 3901685760, 18289152000, 52672757760, 95591301120, 107017666560, 67722117120, 18579456000;
%F Formula
The row polynomials satisfy (1+(6*n-2)*x+4*n^2*x^2)*P(n,x) = (n+(6*n^2+6*n)*x+(4*n^3+20*n^2+12*n)*x^2+(8*n^3+16*n^2+8*n)*x^3)*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[((n + (6*n^2 + 6*n)*x + (4*n^3 + 20*n^2 + 12*n)*x^2 + (8*n^3 + 16*n^2 + 8*n)*x^3)*P[n - 1, x])/(1 + (6*n - 2)*x + 4*n^2*x^2)]];
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] = ((n + (6*n^2 + 6*n)*x + (4*n^3 + 20*n^2 + 12*n)*x^2 + (8*n^3 + 16*n^2 + 8*n)*x^3)*v[i - 1])/(1 + (6*n - 2)*x + 4*n^2*x^2)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.