Real-rooted atlas

A123519 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(108 + 153\,x - 141\,x^{2} - 129\,x^{3} + 12\,x^{4} + 72\,n + 102\,n x - 364\,n x^{2} - 257\,n x^{3} + 44\,n x^{4} - 270\,n^{2} x^{2} - 171\,n^{2} x^{3} + 63\,n^{2} x^{4} - 60\,n^{3} x^{2} - 38\,n^{3} x^{3} + 44\,n^{3} x^{4} + 15\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(252 + 1185\,x + 571\,x^{2} - 765\,x^{3} - 448\,x^{4} + 36\,x^{5} + 144\,n + 870\,n x - 284\,n x^{2} - 1751\,n x^{3} - 666\,n x^{4} + 114\,n x^{5} + 126\,n^{2} x - 528\,n^{2} x^{2} - 1135\,n^{2} x^{3} - 257\,n^{2} x^{4} + 136\,n^{2} x^{5} - 120\,n^{3} x^{2} - 250\,n^{3} x^{3} + 28\,n^{3} x^{4} + 76\,n^{3} x^{5} - 12\,n^{4} x^{3} + 31\,n^{4} x^{4} + 20\,n^{4} x^{5} + 4\,n^{5} x^{4} + 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(54 + 270\,x - 36\,x^{2} - 760\,x^{3} - 588\,x^{4} - 62\,x^{5} + 18\,x^{6} - 504\,n x^{2} - 1128\,n x^{3} - 632\,n x^{4} + 40\,n x^{5} + 48\,n x^{6} - 126\,n^{2} x^{2} - 282\,n^{2} x^{3} - 62\,n^{2} x^{4} + 138\,n^{2} x^{5} + 44\,n^{2} x^{6} + 48\,n^{3} x^{4} + 64\,n^{3} x^{5} + 16\,n^{3} x^{6} + 6\,n^{4} x^{4} + 8\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 56\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0115636103
141304396128869
21531488291619204006877
3571687217964179207600115252079
42131305689986413363286000264963136381827
579531324485190488716807540003525128467281922730505
629681562848257234451985925654000355507212951682539522073619142393
711077123563921230314429068800381076003015129614773696439091272576051200132039571
841340397488885724399615482112023761640022680000013987814455902208139968001996800123904898541663
91542841399463842605428367937498881396174000156138393611621388805846138881967639044249600053278722949126044975341
105757961162379376116468607639465529607827548000100397168648759111424531834470422528627206538240001240279041386086469222440269365077
1121489003655634808512890866019133568000422469960006114082176061204647168435697090562227674700881354240002073037824350355456353034241605632265974247799
128019805126320718802230130520090823838720220952836000356194918656402498380928328842018816197139951360868921856002785584947263217336329628835848830976036864001743590168059
1329930320110514237120959206356004234553164801125388844000199998302054425184305995522323222708224159866441856082619863040031962944921691238105088186547445762585067520217497600838860811354410966289
1411170147534181930220840869530424019440485760005603900404000108848570342401511562223987215542806298624120781752030727165221888000325006263296011194016071682878122024965352213708868014080005284823041893990473504410674929
Technical details
Node id
n-557d67079896dd55 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-90e168c4ffaacb1e

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:      11,       56,        36;
  1:      41,      304,       396,       128;
  2:     153,     1488,      2916,      1920,        400;
  3:     571,     6872,     17964,     17920,       7600,       1152;
  4:    2131,    30568,     99864,    133632,      86000,      26496,       3136;
  5:    7953,   132448,    519048,    871680,     754000,     352512,      84672,      8192;
  6:   29681,   562848,   2572344,   5198592,    5654000,    3555072,    1295168,    253952,     20736;
  7:  110771,  2356392,  12303144,  29068800,   38107600,   30151296,   14773696,   4390912,    725760,    51200;
  8:  413403,  9748888,  57243996, 154821120,  237616400,  226800000,  139878144,  55902208,  13996800,  1996800,  123904;
  9: 1542841, 39946384, 260542836, 793749888, 1396174000, 1561383936, 1162138880, 584613888, 196763904, 42496000, 5327872, 294912;
%F Formula
The row polynomials satisfy (72*n+108+(102*n+153)*x-(60*n^3+270*n^2+364*n+141)*x^2-(38*n^3+171*n^2+257*n+129)*x^3+(2*n^5+15*n^4+44*n^3+63*n^2+44*n+12)*x^4)*P(n,x) = (144*n+252+(126*n^2+870*n+1185)*x-(120*n^3+528*n^2+284*n-571)*x^2-(12*n^4+250*n^3+1135*n^2+1751*n+765)*x^3+(4*n^5+31*n^4+28*n^3-257*n^2-666*n-448)*x^4+(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^5)*P(n-1,x) + (54+270*x-(126*n^2+504*n+36)*x^2-(282*n^2+1128*n+760)*x^3+(6*n^4+48*n^3-62*n^2-632*n-588)*x^4+(8*n^4+64*n^3+138*n^2+40*n-62)*x^5+(2*n^4+16*n^3+44*n^2+48*n+18)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 56*x + 36*x^2.
%t Mathematica
P[0, x_] := 11 + 56*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((144*n + 252 + (126*n^2 + 870*n + 1185)*x - (120*n^3 + 528*n^2 + 284*n - 571)*x^2 - (12*n^4 + 250*n^3 + 1135*n^2 + 1751*n + 765)*x^3 + (4*n^5 + 31*n^4 + 28*n^3 - 257*n^2 - 666*n - 448)*x^4 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^5)*P[n - 1, x] + (54 + 270*x - (126*n^2 + 504*n + 36)*x^2 - (282*n^2 + 1128*n + 760)*x^3 + (6*n^4 + 48*n^3 - 62*n^2 - 632*n - 588)*x^4 + (8*n^4 + 64*n^3 + 138*n^2 + 40*n - 62)*x^5 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^6)*D[P[n - 1, x], x])/(72*n + 108 + (102*n + 153)*x - (60*n^3 + 270*n^2 + 364*n + 141)*x^2 - (38*n^3 + 171*n^2 + 257*n + 129)*x^3 + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 56*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144*n + 252 + (126*n^2 + 870*n + 1185)*x - (120*n^3 + 528*n^2 + 284*n - 571)*x^2 - (12*n^4 + 250*n^3 + 1135*n^2 + 1751*n + 765)*x^3 + (4*n^5 + 31*n^4 + 28*n^3 - 257*n^2 - 666*n - 448)*x^4 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^5)*v[i - 1] + (54 + 270*x - (126*n^2 + 504*n + 36)*x^2 - (282*n^2 + 1128*n + 760)*x^3 + (6*n^4 + 48*n^3 - 62*n^2 - 632*n - 588)*x^4 + (8*n^4 + 64*n^3 + 138*n^2 + 40*n - 62)*x^5 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^6)*deriv(v[i - 1], x))/(72*n + 108 + (102*n + 153)*x - (60*n^3 + 270*n^2 + 364*n + 141)*x^2 - (38*n^3 + 171*n^2 + 257*n + 129)*x^3 + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^4)); 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. [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