Real-rooted atlas

A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).

A007318 → ×(1+(3/2)n+(1/2)n²)

A094305 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A094305
\(P_{n}(x) = \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 6 + 12\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1336
2612624
31030301080
41560906015240
52110521021010521672
628168420560420168281792
73625275612601260756252364608
845360126025203150252012603604511520
9554951980462069306930462019804955528160
106666029707920138601663213860792029706606667584
11788584290128702574036036360362574012870429085878159744
129110926006200204504572072840847207245045200206006109291372736
13105136581903003075075135135180180180180135135750753003081901365105860160
1412016801092043680120120240240360360411840360360240240120120436801092016801201966080

Generated b-files: A094305, A094305 (reversed triangle) ?

Computed in your browser ?

Other recurrences

X ↦ x+1 substitution · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + 3\,x + 3\,x^{2} + x^{3}\right)\,P_{n-3}(x) + \left(-3 - 6\,x - 3\,x^{2}\right)\,P_{n-2}(x) + \left(3 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 6 + 12\,x + 6\,x^{2}\)
Technical details
Node id
n-7f1b09730105c8f0 · revision 3 · JSON
Definition source
oeis_catalog:A094305
derived:shift_x_plus_one (equal by e-545f8764298a874b)
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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,   3;
  2:  6,  12,    6;
  3: 10,  30,   30,   10;
  4: 15,  60,   90,   60,   15;
  5: 21, 105,  210,  210,  105,   21;
  6: 28, 168,  420,  560,  420,  168,   28;
  7: 36, 252,  756, 1260, 1260,  756,  252,   36;
  8: 45, 360, 1260, 2520, 3150, 2520, 1260,  360,  45;
  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 6 + 12*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 6 + 12*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
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 + 3*x; v[3] = 6 + 12*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 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)

Generated with polytool c48629374597.

A094305: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  3,   3;
  2:  6,  12,    6;
  3: 10,  30,   30,   10;
  4: 15,  60,   90,   60,   15;
  5: 21, 105,  210,  210,  105,   21;
  6: 28, 168,  420,  560,  420,  168,   28;
  7: 36, 252,  756, 1260, 1260,  756,  252,   36;
  8: 45, 360, 1260, 2520, 3150, 2520, 1260,  360,  45;
  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F Formula
The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 6 + 12*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 6 + 12*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
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 + 3*x; v[3] = 6 + 12*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); 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.

A094305: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:  1;
  1:  3,   3;
  2:  6,  12,    6;
  3: 10,  30,   30,   10;
  4: 15,  60,   90,   60,   15;
  5: 21, 105,  210,  210,  105,   21;
  6: 28, 168,  420,  560,  420,  168,   28;
  7: 36, 252,  756, 1260, 1260,  756,  252,   36;
  8: 45, 360, 1260, 2520, 3150, 2520, 1260,  360,  45;
  9: 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*x)*P(n-1,x) - (3+6*x+3*x^2)*P(n-2,x) + (1+3*x+3*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 6 + 12*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 6 + 12*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (1 + 3*x + 3*x^2 + x^3)*P[n - 3, x]];
Table[Reverse[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 + 3*x; v[3] = 6 + 12*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (3 + 6*x + 3*x^2)*v[i - 2] + (1 + 3*x + 3*x^2 + x^3)*v[i - 3]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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-6203e3bc70d7618f, e-739fcf83152eb2a2
  • 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-5e33c96e4e906d25, e-6203e3bc70d7618f, e-739fcf83152eb2a2
  • 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-5e33c96e4e906d25, e-6203e3bc70d7618f, e-739fcf83152eb2a2
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5e33c96e4e906d25

Proof chains

4 incoming proof steps. 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 (59)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.