Real-rooted atlas

Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

A088729 (tail) ↗ nonnegative coefficients

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

Recurrence from A088729

\(P_{n}(x) = \left(3 + x\right)\,P_{n-1}(x) + \left(4 + 3\,x\right)\,P_{n-1}'(x) + \left(2\,x\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 13 + 9\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1139123
27579181173
35417652653011602
446838311387066545117575
54729310098959101136501400631222497
654583513624399604982789013843026188413188806
77087261202464451670054558445101012431926104494108150988405
810224756332897247131081959012733190526587575305690719908072301351899222457

All 15 rows

Technical details
Node id
n-4f94f81718ae3273 · revision 0 · JSON
Definition source
oeis_catalog:A088729
Fingerprint
f6-ad16784f8d87fb89

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 1)
  • 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 ?

Row sums ? A075729

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:          13,            9,            1;
   2:          75,           79,           18,           1;
   3:         541,          765,          265,          30,           1;
   4:        4683,         8311,         3870,         665,          45,          1;
   5:       47293,       100989,        59101,       13650,        1400,         63,         1;
   6:      545835,      1362439,       960498,      278901,       38430,       2618,        84,        1;
   7:     7087261,     20246445,     16700545,     5844510,     1012431,      92610,      4494,      108,      1;
   8:   102247563,    328972471,    310819590,   127331905,    26587575,    3056907,    199080,     7230,    135,     1;
   9:  1622632573,   5805917469,   6182314501,  2902305450,   709559950,   98334621,   8047347,   392040,  11055,   165,   1;
  10: 28091567595, 110645911399, 131102078778, 69392377901, 19445781960, 3155560276, 311237388, 19048227, 720225, 16225, 198, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) + (4+3*x)*d/dx P(n-1,x) + 2*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 13 + 9*x + x^2.
%t Mathematica
P[1, x_] := 13 + 9*x + x^2;
P[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] + (4 + 3*x)*D[P[n - 1, x], x] + 2*x*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 13 + 9*x + x^2; for(n = 2, N, my(i = n); v[i] = (3 + x)*v[i - 1] + (4 + 3*x)*deriv(v[i - 1], x) + 2*x*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.