A272866 → basis A013610

OEIS matches ?

• A009973: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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.

Triangle begins:
  0:       1;
  1:       5,       15,         9;
  2:      25,      150,       315,        270,          81;
  3:     125,     1125,      4050,       7425,        7290,         3645,          729;
  4:     625,     7500,     38250,     108000,      184275,       194400,       123930,        43740,          6561;
  5:    3125,    46875,    309375,    1181250,     2885625,      4708125,      5194125,      3827250,       1804275,        492075,         59049;
  6:   15625,   281250,   2278125,   10968750,    34931250,     77456250,    122563125,    139421250,     113177250,      63969750,      23914845,       5314410,        531441;
  7:   78125,  1640625,  15750000,   91546875,   359690625,   1009968750,   2089040625,   3232659375,    3760273125,    3272298750,    2097715725,     961022475,     297606960,      55801305,      4782969;
  8:  390625,  9375000, 104062500,  708750000,  3313406250,  11269125000,  28831950000,  56588625000,   86092621875,  101859525000,   93415518000,   65721537000,   34782813450,   13392313200,   3539397060,    573956280,    43046721;
  9: 1953125, 52734375, 664453125, 5189062500, 28134843750, 112425468750, 343017281250, 816844500000, 1538780034375, 2309984578125, 2769804061875, 2646576180000, 2000476784250, 1180197600750, 531627004350, 176491556100, 40679151345, 5811307335, 387420489;

The row polynomials satisfy P(n,x) = (5+15*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + 15*x + 9*x^2)*P[n - 1, x]];
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] = (5 + 15*x + 9*x^2)*v[i - 1]); 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-707e884996971c57
• 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-707e884996971c57
• 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-707e884996971c57

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

• basis_map A013610 e-707e884996971c57

  Geometric basis map (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]
    • 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.