Real-rooted atlas

A007318 → ⊙ C(n+1,k+1) (shift +2) → reverse

nonnegative coefficients

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

Recurrence
\(\left(3\,n + 10\,n^{2} + 9\,n^{3} + 2\,n^{4}\right)\,P_{n}(x) = \left(6 - 12\,x + 6\,x^{2} + 7\,n - 14\,n x + 7\,n x^{2} - 4\,n^{2} + 8\,n^{2} x - 4\,n^{2} x^{2} - 7\,n^{3} + 14\,n^{3} x - 7\,n^{3} x^{2} - 2\,n^{4} + 4\,n^{4} x - 2\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(12\,n + 22\,n^{2} + 16\,n^{2} x + 16\,n^{3} + 16\,n^{3} x + 4\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8\)
\(P_{1}(x) = 18 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
088
1181230
2326416112
35020015020420
472480720288241584
59898024501960490286006
612817926720896044807683222880
71623024158763175226460907211343687516
8200480033600940801176006720016800160040335920
9242726065340243936426888365904152460290402178441293292
102881056011880057024013305601596672997920316800475202880484992288
1133814872204490122694036808205889312515314824538806134707436037185219315400
1239220384336336246646492492401902700822198176147987845549544112112011211247045674884320
134502730053235046846802147145055212300828184507361640038648610117117001951950163800585060290845350
1451235840815360847974446638592146578432274834560314096640219867648932771842331929632614402329607168641131445440

Generated b-files: A352880 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-4d976aa636e25300 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-4e2cf4a8bda0ea40

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): no

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:   8;
  1:  18,   12;
  2:  32,   64,    16;
  3:  50,  200,   150,     20;
  4:  72,  480,   720,    288,     24;
  5:  98,  980,  2450,   1960,    490,     28;
  6: 128, 1792,  6720,   8960,   4480,    768,     32;
  7: 162, 3024, 15876,  31752,  26460,   9072,   1134,    36;
  8: 200, 4800, 33600,  94080, 117600,  67200,  16800,  1600,   40;
  9: 242, 7260, 65340, 243936, 426888, 365904, 152460, 29040, 2178, 44;
%F Formula
The row polynomials satisfy (2*n^4+9*n^3+10*n^2+3*n)*P(n,x) = -(2*n^4+7*n^3+4*n^2-7*n-6-(4*n^4+14*n^3+8*n^2-14*n-12)*x+(2*n^4+7*n^3+4*n^2-7*n-6)*x^2)*P(n-2,x) + (4*n^4+16*n^3+22*n^2+12*n+(4*n^4+16*n^3+16*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 8, P(1,x) = 18 + 12*x.
%t Mathematica
P[0, x_] := 8;
P[1, x_] := 18 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^4 + 7*n^3 + 4*n^2 - 7*n - 6 - (4*n^4 + 14*n^3 + 8*n^2 - 14*n - 12)*x + (2*n^4 + 7*n^3 + 4*n^2 - 7*n - 6)*x^2)*P[n - 2, x] + (4*n^4 + 16*n^3 + 22*n^2 + 12*n + (4*n^4 + 16*n^3 + 16*n^2)*x)*P[n - 1, x])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; v[2] = 18 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^4 + 7*n^3 + 4*n^2 - 7*n - 6 - (4*n^4 + 14*n^3 + 8*n^2 - 14*n - 12)*x + (2*n^4 + 7*n^3 + 4*n^2 - 7*n - 6)*x^2)*v[i - 2] + (4*n^4 + 16*n^3 + 22*n^2 + 12*n + (4*n^4 + 16*n^3 + 16*n^2)*x)*v[i - 1])/(2*n^4 + 9*n^3 + 10*n^2 + 3*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.

A352880: reversed triangle, offset 2, from the row P0
%e Example
Triangle begins:
   2:  8;
   3: 12,   18;
   4: 16,   64,    32;
   5: 20,  150,   200,     50;
   6: 24,  288,   720,    480,     72;
   7: 28,  490,  1960,   2450,    980,     98;
   8: 32,  768,  4480,   8960,   6720,   1792,    128;
   9: 36, 1134,  9072,  26460,  31752,  15876,   3024,   162;
  10: 40, 1600, 16800,  67200, 117600,  94080,  33600,  4800,  200;
  11: 44, 2178, 29040, 152460, 365904, 426888, 243936, 65340, 7260, 242;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^4-7*n^3+4*n^2+7*n-6)*P(n,x) = -(2*n^4-9*n^3+10*n^2-3*n-(4*n^4-18*n^3+20*n^2-6*n)*x+(2*n^4-9*n^3+10*n^2-3*n)*x^2)*P(n-2,x) + (4*n^4-16*n^3+22*n^2-12*n+(4*n^4-16*n^3+16*n^2)*x)*P(n-1,x) for n >= 4, with P(2,x) = 8, P(3,x) = 18 + 12*x.
%t Mathematica
P[2, x_] := 8;
P[3, x_] := 18 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^4 - 9*n^3 + 10*n^2 - 3*n - (4*n^4 - 18*n^3 + 20*n^2 - 6*n)*x + (2*n^4 - 9*n^3 + 10*n^2 - 3*n)*x^2)*P[n - 2, x] + (4*n^4 - 16*n^3 + 22*n^2 - 12*n + (4*n^4 - 16*n^3 + 16*n^2)*x)*P[n - 1, x])/(2*n^4 - 7*n^3 + 4*n^2 + 7*n - 6)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N - 1)); v[1] = 8; v[2] = 18 + 12*x; for(n = 4, N, my(i = n - 1); v[i] = (-(2*n^4 - 9*n^3 + 10*n^2 - 3*n - (4*n^4 - 18*n^3 + 20*n^2 - 6*n)*x + (2*n^4 - 9*n^3 + 10*n^2 - 3*n)*x^2)*v[i - 2] + (4*n^4 - 16*n^3 + 22*n^2 - 12*n + (4*n^4 - 16*n^3 + 16*n^2)*x)*v[i - 1])/(2*n^4 - 7*n^3 + 4*n^2 + 7*n - 6)); 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(11)

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-8b10d4d15ed56acc
  • 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-8b10d4d15ed56acc
  • 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-8b10d4d15ed56acc

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

Same fingerprint (7)

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.