A272866 → (θ+1)²

OEIS matches ?

No OEIS matches.

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.

Triangle begins:
  0: 1;
  1: 1,  12,    9;
  2: 1,  24,   99,    96,     25;
  3: 1,  36,  270,   720,    750,     324,      49;
  4: 1,  48,  522,  2304,   4875,    5184,    2842,      768,       81;
  5: 1,  60,  855,  5280,  17125,   31428,   33565,    21120,     7695,     1500,      121;
  6: 1,  72, 1269, 10080,  44250,  117288,  195461,   208512,   143370,    63000,    17061,     2592,      169;
  7: 1,  84, 1764, 17136,  95025,  331128,  761117,  1182912,  1258173,   919800,   459921,   154224,    33124,    4116,     225;
  8: 1,  96, 2340, 26880, 180250,  780192, 2299472,  4753920,  7007715,  7428000,  5678288,  3120768,  1218490,  329280,   58500,   6144,   289;
  9: 1, 108, 2997, 39744, 312750, 1619352, 5838546, 15151104, 28858923, 40810500, 43110243, 34089984, 20137026, 8816472, 2814750, 635904, 96237, 8748, 361;

The row polynomials satisfy (1+(9*n-3)*x+(9*n^2+8*n-6)*x^2+(12*n^2-3*n-3)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (1+(9*n+9)*x+(9*n^2+53*n+30)*x^2+(39*n^2+108*n+45)*x^3+(49*n^2+93*n+30)*x^4+(24*n^2+33*n+9)*x^5+(4*n^2+4*n+1)*x^6)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (9*n + 9)*x + (9*n^2 + 53*n + 30)*x^2 + (39*n^2 + 108*n + 45)*x^3 + (49*n^2 + 93*n + 30)*x^4 + (24*n^2 + 33*n + 9)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(1 + (9*n - 3)*x + (9*n^2 + 8*n - 6)*x^2 + (12*n^2 - 3*n - 3)*x^3 + (4*n^2 - 4*n + 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (9*n + 9)*x + (9*n^2 + 53*n + 30)*x^2 + (39*n^2 + 108*n + 45)*x^3 + (49*n^2 + 93*n + 30)*x^4 + (24*n^2 + 33*n + 9)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(1 + (9*n - 3)*x + (9*n^2 + 8*n - 6)*x^2 + (12*n^2 - 3*n - 3)*x^3 + (4*n^2 - 4*n + 1)*x^4)); 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)

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-47750c013c3a531e
• 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-47750c013c3a531e
• 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-47750c013c3a531e

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

• preserver_image euler_polynomial, shift 0 e-47750c013c3a531e

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.