Real-rooted atlas

A139548 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 24\,x + 9\,x^{2} + 36\,n x - 36\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(32 + 192\,x + 432\,x^{2} + 432\,x^{3} + 162\,x^{4} + 288\,n x + 1152\,n x^{2} + 1512\,n x^{3} + 648\,n x^{4} + 288\,n^{2} x^{2} + 864\,n^{2} x^{3} + 648\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1896162266
2641536777613824810031300
3512184321555205529609720008398082857682825000
44096196608232243212386304362880006270566464012032358318088503056222250000
5327681966080298598402123366408709120002257403904384072192042998169603061100160125971200022863772816062500000
626214418874368350355456311427072016422912000567575838721351934115842270303354882693768140802217093120001207207203843918208204857480658561095625000000
7209715217616076838645268484122161971226568622080011477644738563515028701184787038496358413073754710016161404379136001464744740659295081852436484184591943168111995451187213774950720071656250000000

All 15 rows

Technical details
Node id
n-66407c1a4552c33b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d115b0993943a210

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:         8,          96,          162;
  2:        64,        1536,         7776,         13824,           8100;
  3:       512,       18432,       155520,        552960,         972000,          839808,           285768;
  4:      4096,      196608,      2322432,      12386304,       36288000,        62705664,         64012032,         35831808,           8503056;
  5:     32768,     1966080,     29859840,     212336640,      870912000,      2257403904,       3840721920,       4299816960,        3061100160,        1259712000,         228637728;
  6:    262144,    18874368,    350355456,    3114270720,    16422912000,     56757583872,     135193411584,     227030335488,      269376814080,      221709312000,      120720720384,       39182082048,        5748065856;
  7:   2097152,   176160768,   3864526848,   41221619712,   265686220800,   1147764473856,    3515028701184,    7870384963584,    13073754710016,    16140437913600,    14647447406592,     9508185243648,     4184591943168,     1119954511872,      137749507200;
  8:  16777216,  1610612736,  40768634880,  507343011840,  3864526848000,  20033707180032,   74987278958592,  209876932362240,   448243018629120,   737848590336000,   937436634021888,   912785783390208,   669534710906880,   358385443799040,   132239526912000,    30091839012864,    3184768606464;
  9: 134217728, 14495514624, 415840075776, 5914169966592, 51980009472000, 314375097286656, 1390673173413888, 4670715731116032, 12192210106712064, 25086852071424000, 40979372858671104, 53202371374743552, 54634032410001408, 43866378321002496, 26976863490048000, 12277470317248512, 3898156774311936, 771220920950784, 71607703822848;
%F Formula
The row polynomials satisfy (4+(36*n-24)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (32+(288*n+192)*x+(288*n^2+1152*n+432)*x^2+(864*n^2+1512*n+432)*x^3+(648*n^2+648*n+162)*x^4)*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[((32 + (288*n + 192)*x + (288*n^2 + 1152*n + 432)*x^2 + (864*n^2 + 1512*n + 432)*x^3 + (648*n^2 + 648*n + 162)*x^4)*P[n - 1, x])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)]];
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] = ((32 + (288*n + 192)*x + (288*n^2 + 1152*n + 432)*x^2 + (864*n^2 + 1512*n + 432)*x^3 + (648*n^2 + 648*n + 162)*x^4)*v[i - 1])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*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. [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.