Real-rooted atlas

A088996 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 48\,n x + 144\,n x^{2} + 12\,n^{2} + 80\,n^{2} x - 288\,n^{2} x^{2} - 4\,n^{3} - 32\,n^{3} x + 144\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48\,n x^{2} + 64\,n x^{3} - 80\,n^{2} x^{2} - 128\,n^{2} x^{3} + 32\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 6\,n - 16\,n x + 4\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(16\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111617
2380288371
3155443600614410303
4105476844784145408153600348665
59455131261761629009925926400442368013920945
610395655152956462458262528173977600253476864144506880640454043
71351359681216165614688123351859248106944001030997606411538874368528482304033353317503
820270251625155203181267872279244144641338363456003763264389126221398507525615501967362140353331201939158390001
9344594253054655440672453109446783161896963845966188800133356256174082901637296947238799623061504292242315018249512681472000124483151426513
106547290756355196136015525798975361766989638246411546811944320047490453920563212773618144788482253150283104256251949131602329616239128739840004604137832448008743989412454467
1113749310575144982350576038895822022896492514073794560363963588460160017289055371239424552661032853708801207253573650350081783816240268574721709634270265344009612581085904896024108939922636800667032827209958335
1231623414322535985808262880105118484514868814646933843228672120650180880428800649219905882169344240067199315117260862415295314908938241143684281254359859214502921726905548800121404494914592440326045593260020203520135813694897520640054911750272039249385
1379058535806259654219083628003048996069547416046334954509890662442063178967730496002526478319138074214410596714205621730099231908796059762137497669669173260483441459210967901126198820864001216035072343378231296902443019864528388096402902545731724247040819061052305047552004851789604970400914545
1421345804667687527837302727855600944872099231515120155458826935754618881541136684090266688001021346621363058757632478685075432755480576016366114469390528282624414260359876869441454087777967504798462771200010715144498439442241945610540816668821831653785670188929429186392621056283786739651964410265605265392479103877120000457943262900814771606635
Technical details
Node id
n-9f8bb2102c91e8bb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2c953ff2a550d931

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,         16;
  2:        3,         80,         288;
  3:       15,        544,        3600,         6144;
  4:      105,       4768,       44784,       145408,        153600;
  5:      945,      51312,      617616,      2900992,       5926400,        4423680;
  6:    10395,     655152,     9564624,     58262528,     173977600,      253476864,      144506880;
  7:   135135,    9681216,   165614688,   1233518592,    4810694400,    10309976064,    11538874368,     5284823040;
  8:  2027025,  162515520,  3181267872,  27924414464,  133836345600,   376326438912,   622139850752,   561550196736,   214035333120;
  9: 34459425, 3054655440, 67245310944, 678316189696, 3845966188800, 13335625617408, 29016372969472, 38799623061504, 29224231501824, 9512681472000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(32*n^3-80*n^2+48*n)*x-(144*n^3-288*n^2+144*n)*x^2)*P(n-2,x) + ((32*n^3-80*n^2+48*n)*x^2+(64*n^3-128*n^2+64*n)*x^3)*d/dx P(n-2,x) + (4*n^2-6*n+1+(16*n^2-16*n)*x)*P(n-1,x) - (16*n^2-16*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (32*n^3 - 80*n^2 + 48*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*P[n - 2, x] + ((32*n^3 - 80*n^2 + 48*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 6*n + 1 + (16*n^2 - 16*n)*x)*P[n - 1, x] - (16*n^2 - 16*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (32*n^3 - 80*n^2 + 48*n)*x - (144*n^3 - 288*n^2 + 144*n)*x^2)*v[i - 2] + ((32*n^3 - 80*n^2 + 48*n)*x^2 + (64*n^3 - 128*n^2 + 64*n)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 6*n + 1 + (16*n^2 - 16*n)*x)*v[i - 1] - (16*n^2 - 16*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.