Real-rooted atlas

A103633 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(10 + 6\,x - 7\,n - 4\,n x\right)\,P_{n-1}(x) + \left(12 + 14\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21214430
33666408150
414433629211216900
543212961548920272324500
6172860488784677629286726427000
751842160038448378962233678721536128135000
8207369676819699222848016513676160218883584256810000
96220833177678451210794249523685587522179845452879365124050000
1024883214515203801600588672059683204139744198944065408014080017920102424300000
117464964852224143078402526336029678400243558721424780859411201730560335360389122048121500000
122985984209018886693580812966912016924032015678028810570297652260096188044804802560826368860164096729000000
138957952686776322426112005228789767670592008088215046306695043681862401609336325201664012084224191078418432081923645000000
143583180829262643211078000642576738304411399475247694044164140321024273408396813801070085299338241523701763181158445588484014081638421870000000

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(12 + 14\,x + 4\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
Technical details
Node id
n-17ae8aea8bf71051 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-b3c91bcca268a73d)
Fingerprint
f6-a7854389020f6acb

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      2;
  2:    12,     14,      4;
  3:    36,     66,     40,       8;
  4:   144,    336,    292,     112,     16;
  5:   432,   1296,   1548,     920,    272,     32;
  6:  1728,   6048,   8784,    6776,   2928,    672,     64;
  7:  5184,  21600,  38448,   37896,  22336,   7872,   1536,   128;
  8: 20736,  96768, 196992,  228480, 165136,  76160,  21888,  3584,  256;
  9: 62208, 331776, 784512, 1079424, 952368, 558752, 217984, 54528, 7936, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(7*n-10+(4*n-6)*x)*P(n-1,x) + (12+14*x+4*x^2)*d/dx 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[-(7*n - 10 + (4*n - 6)*x)*P[n - 1, x] + (12 + 14*x + 4*x^2)*D[P[n - 1, x], x]];
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] = -(7*n - 10 + (4*n - 6)*x)*v[i - 1] + (12 + 14*x + 4*x^2)*deriv(v[i - 1], x)); 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.