Real-rooted atlas

A034870 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 8\,x + x^{2} + 12\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 32\,x + 24\,x^{2} + 8\,x^{3} + x^{4} + 48\,n x + 64\,n x^{2} + 28\,n x^{3} + 4\,n x^{4} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1416929
21612821612825513
364768216025601500432497533
4256409616128286722800016128548810248199873
5102420480103680245760336000290304164640614401458020001211240029
64096983046082561802240316800036495362897664162201664152017600031944345616914703201
716384458752335462411927552256256003690086437669632281149441556755264064001937936419328615165488225168466797

All 15 rows

Technical details
Node id
n-f4e303fe02b29314 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-86a3c7a2e1168367

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,      16,        9;
  2:     16,     128,      216,       128,         25;
  3:     64,     768,     2160,      2560,       1500,        432,         49;
  4:    256,    4096,    16128,     28672,      28000,      16128,       5488,       1024,         81;
  5:   1024,   20480,   103680,    245760,     336000,     290304,     164640,      61440,      14580,       2000,        121;
  6:   4096,   98304,   608256,   1802240,    3168000,    3649536,    2897664,    1622016,     641520,     176000,      31944,      3456,       169;
  7:  16384,  458752,  3354624,  11927552,   25625600,   36900864,   37669632,   28114944,   15567552,    6406400,    1937936,    419328,     61516,     5488,      225;
  8:  65536, 2097152, 17694720,  73400320,  186368000,  322043904,  401809408,  374865920,  266872320,  146432000,   62013952,  20127744,   4921280,   878080,   108000,    8192,    289;
  9: 262144, 9437184, 90243072, 427819008, 1253376000, 2526806016, 3725869056, 4171235328, 3629463552, 2489344000, 1355447808, 586579968, 200788224, 53738496, 11016000, 1671168, 176868, 11664, 361;
%F Formula
The row polynomials satisfy (4+(12*n-8)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (16+(48*n+32)*x+(16*n^2+64*n+24)*x^2+(16*n^2+28*n+8)*x^3+(4*n^2+4*n+1)*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 + (48*n + 32)*x + (16*n^2 + 64*n + 24)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (4*n^2 + 4*n + 1)*x^4)*P[n - 1, x])/(4 + (12*n - 8)*x + (4*n^2 - 4*n + 1)*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 + (48*n + 32)*x + (16*n^2 + 64*n + 24)*x^2 + (16*n^2 + 28*n + 8)*x^3 + (4*n^2 + 4*n + 1)*x^4)*v[i - 1])/(4 + (12*n - 8)*x + (4*n^2 - 4*n + 1)*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.