Real-rooted atlas

A133214 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 18\,x + 16\,x^{2} + 2\,n + 17\,n x + 24\,n x^{2} + 4\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(20 + 126\,x + 224\,x^{2} + 104\,x^{3} + 14\,n + 135\,n x + 308\,n x^{2} + 164\,n x^{3} + 32\,n^{2} x + 124\,n^{2} x^{2} + 80\,n^{2} x^{3} + 14\,n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2 - 24\,x - 86\,x^{2} - 120\,x^{3} - 56\,x^{4} - 8\,n x - 46\,n x^{2} - 82\,n x^{3} - 44\,n x^{4} - 4\,n^{2} x^{2} - 12\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 32\,x + 12\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
013321257
16322818032503
232115361872768804577
31683100201656010880280019242135
4898964032133740121600516009216448389625
548639403396101883611860807084002136962822410243608295
626572925149447452480105871368187200352665680281681920230433421185
7146256315554628529061768877926484450240475614721565491228016642280965120309404135
8809745395600160367068780710691840803342400560480256243290880636518409227520614400112642862076793
9450467195845858922500936020548869200071919408005994793728324414182411279769602414016002900480016107522457626450155671
102515959693559712256167903984164119544704061416709200595957063683871941427216880320512483449472086507520087724032412876853248244200780001
1114099336192159988464411135924280030207492057650502068760055948472236842480274636822333808640080622127872194278656002957340672256868352103833601146882252364924919
1279238482531306729462087310362583162172574919424402610312104050159436687364361899878336268959503155211756546081283587994009607397700710497149911047316807682569011224576020754653295801
134464238182378849345114047572274824201537229528016031278239319600432972937837444245357436320030069163238400154662146496005737187456000150474035712026907439104030842040320203513856062668800524288191071086526215
142520552366094747161894912307249778188801072584576000002377369280832003620686877982723953301738946563164906274816001875006372096008209563648000026263246929920599860445184094081548288095053414400554434560015099494411141121757508760231425
Technical details
Node id
n-cfbcbd67ce2f033a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6f638175010ea5cb

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:       13,        32,         12;
  1:       63,       228,        180,         32;
  2:      321,      1536,       1872,        768,         80;
  3:     1683,     10020,      16560,      10880,       2800,        192;
  4:     8989,     64032,     133740,     121600,      51600,       9216,        448;
  5:    48639,    403396,    1018836,    1186080,     708400,     213696,      28224,       1024;
  6:   265729,   2514944,    7452480,   10587136,    8187200,    3526656,     802816,      81920,      2304;
  7:  1462563,  15554628,   52906176,   88779264,   84450240,   47561472,   15654912,    2801664,    228096,     5120;
  8:  8097453,  95600160,  367068780,  710691840,  803342400,  560480256,  243290880,   63651840,   9227520,   614400,   11264;
  9: 45046719, 584585892, 2500936020, 5488692000, 7191940800, 5994793728, 3244141824, 1127976960, 241401600, 29004800, 1610752, 24576;
%F Formula
The row polynomials satisfy (2*n+4+(4*n^2+17*n+18)*x+(2*n^3+12*n^2+24*n+16)*x^2)*P(n,x) = (14*n+20+(32*n^2+135*n+126)*x+(14*n^3+124*n^2+308*n+224)*x^2+(12*n^3+80*n^2+164*n+104)*x^3)*P(n-1,x) - (2+(8*n+24)*x+(4*n^2+46*n+86)*x^2+(12*n^2+82*n+120)*x^3+(8*n^2+44*n+56)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 32*x + 12*x^2.
%t Mathematica
P[0, x_] := 13 + 32*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 20 + (32*n^2 + 135*n + 126)*x + (14*n^3 + 124*n^2 + 308*n + 224)*x^2 + (12*n^3 + 80*n^2 + 164*n + 104)*x^3)*P[n - 1, x] - (2 + (8*n + 24)*x + (4*n^2 + 46*n + 86)*x^2 + (12*n^2 + 82*n + 120)*x^3 + (8*n^2 + 44*n + 56)*x^4)*D[P[n - 1, x], x])/(2*n + 4 + (4*n^2 + 17*n + 18)*x + (2*n^3 + 12*n^2 + 24*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 32*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 20 + (32*n^2 + 135*n + 126)*x + (14*n^3 + 124*n^2 + 308*n + 224)*x^2 + (12*n^3 + 80*n^2 + 164*n + 104)*x^3)*v[i - 1] - (2 + (8*n + 24)*x + (4*n^2 + 46*n + 86)*x^2 + (12*n^2 + 82*n + 120)*x^3 + (8*n^2 + 44*n + 56)*x^4)*deriv(v[i - 1], x))/(2*n + 4 + (4*n^2 + 17*n + 18)*x + (2*n^3 + 12*n^2 + 24*n + 16)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023