Real-rooted atlas

A347056: Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.

A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n²

A347056 nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A347056
\(\left(n\right)\,P_{n}(x) = \left(1 + 3\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-1 - x - n - n x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
26161032
310405020120
4158015012035400
521140350420245561232
6282247001120980448843584
73633612602520294020167561209984

Show all 15 stored rows

Generated b-files: A001794 (row sums), A347056 ?

Computed in your browser ?

Technical details
Node id
n-32779d2065847504 · revision 1 · JSON
Definition source
oeis_catalog:A347056
Fingerprint
f6-4e3d4a8be3fd75f4

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  3,   4;
  2:  6,  16,   10;
  3: 10,  40,   50,   20;
  4: 15,  80,  150,  120,    35;
  5: 21, 140,  350,  420,   245,    56;
  6: 28, 224,  700, 1120,   980,   448,    84;
  7: 36, 336, 1260, 2520,  2940,  2016,   756,  120;
  8: 45, 480, 2100, 5040,  7350,  6720,  3780, 1200,  165;
  9: 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+1+(n+3)*x)*P(n-1,x) - (n+1+(n+1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (n + 3)*x)*P[n - 1, x] - (n + 1 + (n + 1)*x)*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (n + 3)*x)*v[i - 1] - (n + 1 + (n + 1)*x)*v[i - 2])/(n)); 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)

Generated with polytool c48629374597.

A347056: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  3,   4;
  2:  6,  16,   10;
  3: 10,  40,   50,   20;
  4: 15,  80,  150,  120,    35;
  5: 21, 140,  350,  420,   245,    56;
  6: 28, 224,  700, 1120,   980,   448,    84;
  7: 36, 336, 1260, 2520,  2940,  2016,   756,  120;
  8: 45, 480, 2100, 5040,  7350,  6720,  3780, 1200,  165;
  9: 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+1+(n+3)*x)*P(n-1,x) - (n+1+(n+1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (n + 3)*x)*P[n - 1, x] - (n + 1 + (n + 1)*x)*P[n - 2, x])/(n)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (n + 3)*x)*v[i - 1] - (n + 1 + (n + 1)*x)*v[i - 2])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6ba022abf95075ed
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-6ba022abf95075ed
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-6ba022abf95075ed

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 7 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  4. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  5. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.