Real-rooted atlas

A123519 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-18 - 30\,x + 24\,x^{2} - 12\,n - 20\,n x + 52\,n x^{2} + 36\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-42 - 190\,x - 104\,x^{2} + 96\,x^{3} - 24\,n - 124\,n x + 36\,n x^{2} + 176\,n x^{3} - 12\,n^{2} x + 76\,n^{2} x^{2} + 96\,n^{2} x^{3} + 16\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-9 - 54\,x - 56\,x^{2} + 56\,x^{3} + 48\,x^{4} + 48\,n x^{2} + 128\,n x^{3} + 64\,n x^{4} + 12\,n^{2} x^{2} + 32\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 56\,x + 48\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0115648115
1413045282561129
2153148838883840128010649
35716872239523584024320614497699
421313056813315226726427520014131228672878299
579531324486920641743360241280018800647741441310727773905
629681562848342979210397184180928001896038411841536406323258982467967281
71107712356392164041925813760012194432016080691213507379270254592206438402621440588353851
841340397488887632532830964224076037248012096000001278885888894435328398131200102236160115343365051325251
9154284139946384347390448158749977644677568008327380992106252697609353822208559683993621757952004959764485033164843069552441
10575796116237937615529147687893105920250481536005354515660880083304448850935152646408142848033475788800115458703362365587456218103808365071066825
112148900365563480868385448803826713600013519038720032608438272055958534553669711534489663364969267241653370880019298097561659793997824111232942089395240963078779458259
1280198051263207188029735073600181647677440707049075200189970623283236799851970565261472301056560753639424044488799027202593126350848107890920652830338239692851673825280402653184025849842435499
13299303201105142371201278941808008469106329603601244300800106665761095682302565119590437171563331584454731212390404230136987648029754595999744155713032683525877679521792151263379456023756537856017179869184216196102239649
1411170147534181930220854492707232038880971520001793248129280058052570849280138199974764544248684900777984343556983554048366859360665600302551285104640191044540956672906829831864323131809621606474289512448001082331758592730144440321801943435356769
Technical details
Node id
n-dfde18ed92c08794 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b3700e8c935da743

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,        48;
  1:      41,      304,       528,        256;
  2:     153,     1488,      3888,       3840,       1280;
  3:     571,     6872,     23952,      35840,      24320,       6144;
  4:    2131,    30568,    133152,     267264,     275200,     141312,       28672;
  5:    7953,   132448,    692064,    1743360,    2412800,    1880064,      774144,     131072;
  6:   29681,   562848,   3429792,   10397184,   18092800,   18960384,    11841536,    4063232,     589824;
  7:  110771,  2356392,  16404192,   58137600,  121944320,  160806912,   135073792,   70254592,   20643840,    2621440;
  8:  413403,  9748888,  76325328,  309642240,  760372480, 1209600000,  1278885888,  894435328,  398131200,  102236160,  11534336;
  9: 1542841, 39946384, 347390448, 1587499776, 4467756800, 8327380992, 10625269760, 9353822208, 5596839936, 2175795200, 495976448, 50331648;
%F Formula
The row polynomials satisfy -(12*n+18+(20*n+30)*x-(8*n^3+36*n^2+52*n+24)*x^2)*P(n,x) = -(24*n+42+(12*n^2+124*n+190)*x-(16*n^3+76*n^2+36*n-104)*x^2-(16*n^3+96*n^2+176*n+96)*x^3)*P(n-1,x) - (9+54*x-(12*n^2+48*n-56)*x^2-(32*n^2+128*n+56)*x^3-(16*n^2+64*n+48)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 56*x + 48*x^2.
%t Mathematica
P[0, x_] := 11 + 56*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n + 42 + (12*n^2 + 124*n + 190)*x - (16*n^3 + 76*n^2 + 36*n - 104)*x^2 - (16*n^3 + 96*n^2 + 176*n + 96)*x^3)*P[n - 1, x] - (9 + 54*x - (12*n^2 + 48*n - 56)*x^2 - (32*n^2 + 128*n + 56)*x^3 - (16*n^2 + 64*n + 48)*x^4)*D[P[n - 1, x], x])/(-12*n - 18 - (20*n + 30)*x + (8*n^3 + 36*n^2 + 52*n + 24)*x^2)]];
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 + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n + 42 + (12*n^2 + 124*n + 190)*x - (16*n^3 + 76*n^2 + 36*n - 104)*x^2 - (16*n^3 + 96*n^2 + 176*n + 96)*x^3)*v[i - 1] - (9 + 54*x - (12*n^2 + 48*n - 56)*x^2 - (32*n^2 + 128*n + 56)*x^3 - (16*n^2 + 64*n + 48)*x^4)*deriv(v[i - 1], x))/(-12*n - 18 - (20*n + 30)*x + (8*n^3 + 36*n^2 + 52*n + 24)*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

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