Real-rooted atlas

A121574 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-78\,n + 189\,n x + 972\,n x^{2} + 405\,n x^{3} - 117\,n^{2} x + 459\,n^{2} x^{2} + 648\,n^{2} x^{3} - 81\,n^{3} x^{2} + 243\,n^{3} x^{3}\right)\,P_{n}(x) = \left(390 + 2916\,x + 7236\,x^{2} + 6804\,x^{3} + 2430\,x^{4} - 624\,n - 1881\,n x + 1188\,n x^{2} + 6561\,n x^{3} + 3888\,n x^{4} - 1287\,n^{2} x - 3294\,n^{2} x^{2} - 405\,n^{2} x^{3} + 1458\,n^{2} x^{4} - 648\,n^{3} x^{2} - 972\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(169 + 1599\,x + 5598\,x^{2} + 8802\,x^{3} + 5913\,x^{4} + 1215\,x^{5} + 507\,n x + 3627\,n x^{2} + 8829\,n x^{3} + 8181\,n x^{4} + 1944\,n x^{5} + 351\,n^{2} x^{2} + 1863\,n^{2} x^{3} + 2673\,n^{2} x^{4} + 729\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
210422779
333210297108648
4109930205216204055116
5360385211583143647695145839312
6118915300583209925283430335345103296128
7392759046272727591732684045430110137781174962196864
812970223134121105831986364726755408677420412005423765904916101952
94283782983051757111612278029078190321693122207047591154162066715196830116868096
1014148130474662147652977114052164416635222292512195148926110557224388542427676370649539841374976
1146728011078136870531303540738968719662151395628218149813363710855393205225246011596291302793017721257646016149504
121543321399344403462011281573454376439727008581401554261035227239291727154005643253881235782657063654822099910908690873342767994880
1350972431429450621355213106680745405621292007490447853451766593019356769233649048520738959242778570378010245178647247651506035234538322320522302507864064
1416835050508613010523479820528801473480997016991752349775902243933821052004784135458324264864216652781054372601308457343164314238038094373292781227628710717445352130353336320
Technical details
Node id
n-a7b6f6be27bd465f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d31923a2b9566825

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:     3,      6;
  2:    10,     42,      27;
  3:    33,    210,     297,      108;
  4:   109,    930,    2052,     1620,      405;
  5:   360,   3852,   11583,    14364,     7695,     1458;
  6:  1189,  15300,   58320,    99252,    83430,    33534,     5103;
  7:  3927,  59046,  272727,   591732,   684045,   430110,   137781,   17496;
  8: 12970, 223134, 1211058,  3198636,  4726755,  4086774,  2041200,  542376,   59049;
  9: 42837, 829830, 5175711, 16122780, 29078190, 32169312, 22070475, 9115416, 2066715, 196830;
%F Formula
The row polynomials satisfy -(78*n+(117*n^2-189*n)*x+(81*n^3-459*n^2-972*n)*x^2-(243*n^3+648*n^2+405*n)*x^3)*P(n,x) = -(624*n-390+(1287*n^2+1881*n-2916)*x+(648*n^3+3294*n^2-1188*n-7236)*x^2+(972*n^3+405*n^2-6561*n-6804)*x^3-(1458*n^2+3888*n+2430)*x^4)*P(n-1,x) + (169+(507*n+1599)*x+(351*n^2+3627*n+5598)*x^2+(1863*n^2+8829*n+8802)*x^3+(2673*n^2+8181*n+5913)*x^4+(729*n^2+1944*n+1215)*x^5)*d/dx 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[(-(624*n - 390 + (1287*n^2 + 1881*n - 2916)*x + (648*n^3 + 3294*n^2 - 1188*n - 7236)*x^2 + (972*n^3 + 405*n^2 - 6561*n - 6804)*x^3 - (1458*n^2 + 3888*n + 2430)*x^4)*P[n - 1, x] + (169 + (507*n + 1599)*x + (351*n^2 + 3627*n + 5598)*x^2 + (1863*n^2 + 8829*n + 8802)*x^3 + (2673*n^2 + 8181*n + 5913)*x^4 + (729*n^2 + 1944*n + 1215)*x^5)*D[P[n - 1, x], x])/(-78*n - (117*n^2 - 189*n)*x - (81*n^3 - 459*n^2 - 972*n)*x^2 + (243*n^3 + 648*n^2 + 405*n)*x^3)]];
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] = (-(624*n - 390 + (1287*n^2 + 1881*n - 2916)*x + (648*n^3 + 3294*n^2 - 1188*n - 7236)*x^2 + (972*n^3 + 405*n^2 - 6561*n - 6804)*x^3 - (1458*n^2 + 3888*n + 2430)*x^4)*v[i - 1] + (169 + (507*n + 1599)*x + (351*n^2 + 3627*n + 5598)*x^2 + (1863*n^2 + 8829*n + 8802)*x^3 + (2673*n^2 + 8181*n + 5913)*x^4 + (729*n^2 + 1944*n + 1215)*x^5)*deriv(v[i - 1], x))/(-78*n - (117*n^2 - 189*n)*x - (81*n^3 - 459*n^2 - 972*n)*x^2 + (243*n^3 + 648*n^2 + 405*n)*x^3)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.