Real-rooted atlas

A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).

A007318 → ⊙ C(n+1,k+1) (shift +2)

A352880 nonnegative coefficients

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

Recurrence from OEIS A352880
\(\left(9\,n + 3\,n^{2}\right)\,P_{n}(x) = \left(8\,n + 26\,n x + 8\,n^{2} + 5\,n^{2} x + 2\,n^{3} - 4\,n^{3} x\right)\,P_{n-1}(x) + \left(2 - 8\,x + 8\,x^{2} + 7\,n + 2\,n x - 11\,n x^{2} - 5\,n^{2} + 2\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} + 4\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n + 26\,n x - 28\,n x^{2} + 10\,n x^{3} + 3\,n^{2} x - 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + 2\,n^{3} - 8\,n^{3} x + 10\,n^{3} x^{2} - 4\,n^{3} x^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 8\)
\(P_{1}(x) = 12 + 18\,x\)
\(P_{2}(x) = 16 + 64\,x + 32\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
088
1121830
2166432112
32015020050420
424288720480721584
52849019602450980986006
632768448089606720179212822880
73611349072264603175215876302416287516
8401600168006720011760094080336004800200335920
9442178290401524603659044268882439366534072602421293292
104828804752031680099792015966721330560570240118800105602884992288
1152371874360613470245388051531485889312368082012269402044901487233819315400
1256470411211211211205549544147987842219817619027008924924024664643363362038439274884320
136058501638001951950117117003864861073616400828184505521230021471450468468053235027300450290845350
1464716823296032614402331929693277184219867648314096640274834560146578432466385928479744815360358405121131445440

Generated b-files: A352880 ?

Computed in your browser ?

Technical details
Node id
n-7a4827a398e28fe4 · revision 1 · JSON
Definition source
oeis_catalog:A352880
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: 12,   18;
  2: 16,   64,    32;
  3: 20,  150,   200,     50;
  4: 24,  288,   720,    480,     72;
  5: 28,  490,  1960,   2450,    980,     98;
  6: 32,  768,  4480,   8960,   6720,   1792,    128;
  7: 36, 1134,  9072,  26460,  31752,  15876,   3024,   162;
  8: 40, 1600, 16800,  67200, 117600,  94080,  33600,  4800,  200;
  9: 44, 2178, 29040, 152460, 365904, 426888, 243936, 65340, 7260, 242;
%F Formula
The row polynomials satisfy (3*n^2+9*n)*P(n,x) = (2*n^3+8*n^2+8*n-(4*n^3-5*n^2-26*n)*x)*P(n-1,x) - (4*n^3+5*n^2-7*n-2-(4*n^3+2*n^2+2*n-8)*x-(8*n^3-5*n^2-11*n+8)*x^2)*P(n-2,x) + (2*n^3-8*n-(8*n^3-3*n^2-26*n)*x+(10*n^3-6*n^2-28*n)*x^2-(4*n^3-3*n^2-10*n)*x^3)*P(n-3,x) for n >= 3, with P(0,x) = 8, P(1,x) = 12 + 18*x, P(2,x) = 16 + 64*x + 32*x^2.
%t Mathematica
P[0, x_] := 8;
P[1, x_] := 12 + 18*x;
P[2, x_] := 16 + 64*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 8*n^2 + 8*n - (4*n^3 - 5*n^2 - 26*n)*x)*P[n - 1, x] - (4*n^3 + 5*n^2 - 7*n - 2 - (4*n^3 + 2*n^2 + 2*n - 8)*x - (8*n^3 - 5*n^2 - 11*n + 8)*x^2)*P[n - 2, x] + (2*n^3 - 8*n - (8*n^3 - 3*n^2 - 26*n)*x + (10*n^3 - 6*n^2 - 28*n)*x^2 - (4*n^3 - 3*n^2 - 10*n)*x^3)*P[n - 3, x])/(3*n^2 + 9*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] = 12 + 18*x; v[3] = 16 + 64*x + 32*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 + 8*n^2 + 8*n - (4*n^3 - 5*n^2 - 26*n)*x)*v[i - 1] - (4*n^3 + 5*n^2 - 7*n - 2 - (4*n^3 + 2*n^2 + 2*n - 8)*x - (8*n^3 - 5*n^2 - 11*n + 8)*x^2)*v[i - 2] + (2*n^3 - 8*n - (8*n^3 - 3*n^2 - 26*n)*x + (10*n^3 - 6*n^2 - 28*n)*x^2 - (4*n^3 - 3*n^2 - 10*n)*x^3)*v[i - 3])/(3*n^2 + 9*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: 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
The row polynomials satisfy (3*n^2-3*n-6)*P(n,x) = (2*n^3-4*n^2-(4*n^3-29*n^2+42*n)*x)*P(n-1,x) - (4*n^3-19*n^2+21*n-(4*n^3-22*n^2+42*n-36)*x-(8*n^3-53*n^2+105*n-54)*x^2)*P(n-2,x) + (2*n^3-12*n^2+16*n-(8*n^3-51*n^2+82*n-24)*x+(10*n^3-66*n^2+116*n-48)*x^2-(4*n^3-27*n^2+50*n-24)*x^3)*P(n-3,x) for n >= 5, with P(2,x) = 8, P(3,x) = 12 + 18*x, P(4,x) = 16 + 64*x + 32*x^2.
%t Mathematica
P[2, x_] := 8;
P[3, x_] := 12 + 18*x;
P[4, x_] := 16 + 64*x + 32*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 4*n^2 - (4*n^3 - 29*n^2 + 42*n)*x)*P[n - 1, x] - (4*n^3 - 19*n^2 + 21*n - (4*n^3 - 22*n^2 + 42*n - 36)*x - (8*n^3 - 53*n^2 + 105*n - 54)*x^2)*P[n - 2, x] + (2*n^3 - 12*n^2 + 16*n - (8*n^3 - 51*n^2 + 82*n - 24)*x + (10*n^3 - 66*n^2 + 116*n - 48)*x^2 - (4*n^3 - 27*n^2 + 50*n - 24)*x^3)*P[n - 3, x])/(3*n^2 - 3*n - 6)]];
Table[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] = 12 + 18*x; v[3] = 16 + 64*x + 32*x^2; for(n = 5, N, my(i = n - 1); v[i] = ((2*n^3 - 4*n^2 - (4*n^3 - 29*n^2 + 42*n)*x)*v[i - 1] - (4*n^3 - 19*n^2 + 21*n - (4*n^3 - 22*n^2 + 42*n - 36)*x - (8*n^3 - 53*n^2 + 105*n - 54)*x^2)*v[i - 2] + (2*n^3 - 12*n^2 + 16*n - (8*n^3 - 51*n^2 + 82*n - 24)*x + (10*n^3 - 66*n^2 + 116*n - 48)*x^2 - (4*n^3 - 27*n^2 + 50*n - 24)*x^3)*v[i - 3])/(3*n^2 - 3*n - 6)); 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(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-2d36970d4b706ae4
  • 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-2d36970d4b706ae4
  • 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-2d36970d4b706ae4

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 3 proof steps

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.