A272866 → D+1

OEIS matches ?

• A053464: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:  4,   5,    1;
  2:  7,  28,   29,    10,      1;
  3: 10,  69,  165,   165,     75,     15,       1;
  4: 13, 128,  490,   924,    915,    492,     142,      20,       1;
  5: 16, 205, 1085,  3070,   5050,   4983,    2995,    1090,     230,      25,       1;
  6: 19, 300, 2031,  7710,  18060,  27192,   26795,   17418,    7440,    2040,     339,      30,      1;
  7: 22, 413, 3409, 16275,  49791, 102396,  144914,  142747,   98315,   47208,   15582,    3423,    469,     35,     1;
  8: 25, 544, 5300, 30520, 115570, 303240,  566888,  766400,  755035,  543560,  285320,  108192,  29050,   5320,   620,   40,   1;
  9: 28, 693, 7785, 52524, 237420, 759906, 1776306, 3087000, 4029228, 3970935, 2960379, 1666584, 703920, 220122, 49770, 7812, 792, 45, 1;

The row polynomials satisfy (3*n-2+(2*n+1)*x+x^2)*P(n,x) = (3*n+1+(11*n+6)*x+(9*n+11)*x^2+(2*n+6)*x^3+x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 1 + (11*n + 6)*x + (9*n + 11)*x^2 + (2*n + 6)*x^3 + x^4)*P[n - 1, x])/(3*n - 2 + (2*n + 1)*x + x^2)]];
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] = ((3*n + 1 + (11*n + 6)*x + (9*n + 11)*x^2 + (2*n + 6)*x^3 + x^4)*v[i - 1])/(3*n - 2 + (2*n + 1)*x + x^2)); 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-5c9a5bac3deec721
• 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-5c9a5bac3deec721
• 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-5c9a5bac3deec721

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 derivative_polynomial, shift 0 e-5c9a5bac3deec721

  Laguerre’s theorem on differential operators [SP14] 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.