Real-rooted atlas

A034870 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 24\,x + 9\,x^{2} + 36\,n x - 36\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 96\,x + 216\,x^{2} + 216\,x^{3} + 81\,x^{4} + 144\,n x + 576\,n x^{2} + 756\,n x^{3} + 324\,n x^{4} + 144\,n^{2} x^{2} + 432\,n^{2} x^{3} + 324\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
144881133
2163841944345620257825
3642304194406912012150010497635721353125
425612288145152774144226800039191044000752223948853144113890625
51024614409331206635520272160007054387212002256013436928095659380393660007144929501953125
64096294912547430448660480256608000886837248211239705635473489924209012720346420800018862612566122200328981352917119140625
716384137625630191616322043904207567360089669099522746116172861487382528102138708672126097171200114433182864742826972163269212455687496446241076168025559814453125

All 15 rows

Technical details
Node id
n-a43d8d44a6eb5699 · 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:      4,       48,        81;
  2:     16,      384,      1944,        3456,         2025;
  3:     64,     2304,     19440,       69120,       121500,       104976,         35721;
  4:    256,    12288,    145152,      774144,      2268000,      3919104,       4000752,       2239488,         531441;
  5:   1024,    61440,    933120,     6635520,     27216000,     70543872,     120022560,     134369280,       95659380,       39366000,        7144929;
  6:   4096,   294912,   5474304,    48660480,    256608000,    886837248,    2112397056,    3547348992,     4209012720,     3464208000,     1886261256,       612220032,        89813529;
  7:  16384,  1376256,  30191616,   322043904,   2075673600,   8966909952,   27461161728,   61487382528,   102138708672,   126097171200,   114433182864,     74282697216,     32692124556,     8749644624,     1076168025;
  8:  65536,  6291456, 159252480,  1981808640,  15095808000,  78256668672,  292919058432,  819831767040,  1750949291520,  2882221056000,  3661861851648,   3565569466368,   2615369964480,  1399943139840,   516560652000,   117546246144,   12440502369;
  9: 262144, 28311552, 812187648, 11551113216, 101523456000, 614013861888, 2716158541824, 9122491662336, 23812910364672, 48997757952000, 80037837614592, 103910881591296, 106707094550784, 85676520158208, 52689186504000, 23979434213376, 7613587449828, 1506290861232, 139858796529;
%F Formula
The row polynomials satisfy (4+(36*n-24)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (16+(144*n+96)*x+(144*n^2+576*n+216)*x^2+(432*n^2+756*n+216)*x^3+(324*n^2+324*n+81)*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[((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*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] = ((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*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.