Real-rooted atlas

A272866 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 4\,x^{2} + 10\,n x + 8\,n x^{2}\right)\,P_{n}(x) = \left(25 + 100\,x + 140\,x^{2} + 80\,x^{3} + 16\,x^{4} + 50\,n x + 140\,n x^{2} + 120\,n x^{3} + 32\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15201237
225200420320801045
31251500540088007200288044826353
4625100005100012800018200015360076160204802304624169
5312562500412500140000028500003720000319200017920006336001280001126414204989
615625375000303750013000000345000006120000075320000652800003974400016640000456192073728053248314464573
7781252187500210000001085000003552500007980000001283800000151360000013204800008512000004001536001333248002981888040140802457606821652745

All 15 rows

Technical details
Node id
n-515d78eac02765ac · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0136bddfb709905a

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:       5,       20,        12;
  2:      25,      200,       420,        320,          80;
  3:     125,     1500,      5400,       8800,        7200,        2880,          448;
  4:     625,    10000,     51000,     128000,      182000,      153600,        76160,        20480,         2304;
  5:    3125,    62500,    412500,    1400000,     2850000,     3720000,      3192000,      1792000,       633600,       128000,        11264;
  6:   15625,   375000,   3037500,   13000000,    34500000,    61200000,     75320000,     65280000,     39744000,     16640000,      4561920,       737280,        53248;
  7:   78125,  2187500,  21000000,  108500000,   355250000,   798000000,   1283800000,   1513600000,   1320480000,    851200000,    400153600,    133324800,     29818880,     4014080,      245760;
  8:  390625, 12500000, 138750000,  840000000,  3272500000,  8904000000,  17718400000,  26496000000,  30232800000,  26496000000,  17819648000,   9117696000,   3485081600,   963379200,   181862400,   20971520,    1114112;
  9: 1953125, 70312500, 885937500, 6150000000, 27787500000, 88830000000, 210798000000, 382464000000, 540367200000, 600880000000, 528359040000, 367165440000, 200438784000, 84897792000, 27316224000, 6448742400, 1052835840, 106168320, 4980736;
%F Formula
The row polynomials satisfy (5+10*n*x+(8*n-4)*x^2)*P(n,x) = (25+(50*n+100)*x+(140*n+140)*x^2+(120*n+80)*x^3+(32*n+16)*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[((25 + (50*n + 100)*x + (140*n + 140)*x^2 + (120*n + 80)*x^3 + (32*n + 16)*x^4)*P[n - 1, x])/(5 + 10*n*x + (8*n - 4)*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] = ((25 + (50*n + 100)*x + (140*n + 140)*x^2 + (120*n + 80)*x^3 + (32*n + 16)*x^4)*v[i - 1])/(5 + 10*n*x + (8*n - 4)*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

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.