Real-rooted atlas

A064861 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n}(x) = \left(22 + 139\,x + 345\,x^{2} + 421\,x^{3} + 253\,x^{4} + 60\,x^{5} + 19\,n + 125\,n x + 321\,n x^{2} + 403\,n x^{3} + 248\,n x^{4} + 60\,n x^{5}\right)\,P_{n-1}(x) + \left(-12 - 98\,x - 328\,x^{2} - 576\,x^{3} - 560\,x^{4} - 286\,x^{5} - 60\,x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 6 + 19\,x + 15\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
06191540
1188714075320
2362285415702251600
3108864276344153525112512800
421620528118171192029512825337564000
5648723634614919471464801399507425016875512000
612961641690936287736568801719340568350256500506252560000
738885572835488813178883145083500202553017503611250143437525312520480000

All 15 rows

Technical details
Node id
n-b70b11f2df74aed6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b3ba2a8274ccf0a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     6,     19,      15;
  1:    18,     87,     140,       75;
  2:    36,    228,     541,      570,      225;
  3:   108,    864,    2763,     4415,     3525,      1125;
  4:   216,   2052,    8118,    17119,    20295,     12825,      3375;
  5:   648,   7236,   34614,    91947,   146480,    139950,     74250,     16875;
  6:  1296,  16416,   90936,   287736,   568801,    719340,    568350,    256500,     50625;
  7:  3888,  55728,  354888,  1317888,  3145083,   5002025,   5301750,   3611250,   1434375,   253125;
  8:  7776, 123120,  876960,  3700440, 10243830,  19439299,  25609575,  23127750,  13702500,  4809375,   759375;
  9: 23328, 408240, 3246480, 15486120, 49233690, 109537047, 174025220, 197431125, 156746250, 82940625, 26325000, 3796875;
%F Formula
The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (19*n+22+(125*n+139)*x+(321*n+345)*x^2+(403*n+421)*x^3+(248*n+253)*x^4+(60*n+60)*x^5)*P(n-1,x) - (12+98*x+328*x^2+576*x^3+560*x^4+286*x^5+60*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 19*x + 15*x^2.
%t Mathematica
P[0, x_] := 6 + 19*x + 15*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 22 + (125*n + 139)*x + (321*n + 345)*x^2 + (403*n + 421)*x^3 + (248*n + 253)*x^4 + (60*n + 60)*x^5)*P[n - 1, x] - (12 + 98*x + 328*x^2 + 576*x^3 + 560*x^4 + 286*x^5 + 60*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 6 + 19*x + 15*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 22 + (125*n + 139)*x + (321*n + 345)*x^2 + (403*n + 421)*x^3 + (248*n + 253)*x^4 + (60*n + 60)*x^5)*v[i - 1] - (12 + 98*x + 328*x^2 + 576*x^3 + 560*x^4 + 286*x^5 + 60*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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.