Real-rooted atlas

A159854 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 15\,x + 48\,x^{2} + 48\,x^{3} + 16\,x^{4} + 3\,n x + 20\,n x^{2} + 36\,n x^{3} + 16\,n x^{4} + n^{2} x^{2} + 4\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n}(x) = \left(2 + 38\,x + 174\,x^{2} + 314\,x^{3} + 248\,x^{4} + 72\,x^{5} + 6\,n x + 50\,n x^{2} + 132\,n x^{3} + 136\,n x^{4} + 48\,n x^{5} + 2\,n^{2} x^{2} + 10\,n^{2} x^{3} + 16\,n^{2} x^{4} + 8\,n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 16\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01163653
1240144128314
24964687684001736
3822413683200320011529152
4165123744112641640011520313646592
53211529792358406800070272376328192230912
664256024768106496248000336384266560114688207361120256
712856326105630105682880013870081448832925696331776512005341184

All 15 rows

Technical details
Node id
n-9305fa2263e02428 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8d0a5745cf4269c7

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,    16,     36;
  1:   2,    40,    144,     128;
  2:   4,    96,    468,     768,     400;
  3:   8,   224,   1368,    3200,    3200,     1152;
  4:  16,   512,   3744,   11264,   16400,    11520,     3136;
  5:  32,  1152,   9792,   35840,   68000,    70272,    37632,     8192;
  6:  64,  2560,  24768,  106496,  248000,   336384,   266560,   114688,    20736;
  7: 128,  5632,  61056,  301056,  828800,  1387008,  1448832,   925696,   331776,   51200;
  8: 256, 12288, 147456,  819200, 2598400,  5160960,  6673408,  5636096,  3006720,  921600,  123904;
  9: 512, 26624, 350208, 2162688, 7756800, 17805312, 27396096, 28704768, 20279808, 9267200, 2478080, 294912;
%F Formula
The row polynomials satisfy (1+(3*n+15)*x+(n^2+20*n+48)*x^2+(4*n^2+36*n+48)*x^3+(4*n^2+16*n+16)*x^4)*P(n,x) = (2+(6*n+38)*x+(2*n^2+50*n+174)*x^2+(10*n^2+132*n+314)*x^3+(16*n^2+136*n+248)*x^4+(8*n^2+48*n+72)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 36*x^2.
%t Mathematica
P[0, x_] := 1 + 16*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2 + (6*n + 38)*x + (2*n^2 + 50*n + 174)*x^2 + (10*n^2 + 132*n + 314)*x^3 + (16*n^2 + 136*n + 248)*x^4 + (8*n^2 + 48*n + 72)*x^5)*P[n - 1, x])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (6*n + 38)*x + (2*n^2 + 50*n + 174)*x^2 + (10*n^2 + 132*n + 314)*x^3 + (16*n^2 + 136*n + 248)*x^4 + (8*n^2 + 48*n + 72)*x^5)*v[i - 1])/(1 + (3*n + 15)*x + (n^2 + 20*n + 48)*x^2 + (4*n^2 + 36*n + 48)*x^3 + (4*n^2 + 16*n + 16)*x^4)); 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.