A272866 → basis A007318

OEIS matches ?

• A001020: 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,        5,        1;
  2:      25,       50,       35,        10,         1;
  3:     125,      375,      450,       275,        90,        15,         1;
  4:     625,     2500,     4250,      4000,      2275,       800,       170,        20,         1;
  5:    3125,    15625,    34375,     43750,     35625,     19375,      7125,      1750,       275,        25,        1;
  6:   15625,    93750,   253125,    406250,    431250,    318750,    168125,     63750,     17250,      3250,      405,       30,       1;
  7:   78125,   546875,  1750000,   3390625,   4440625,   4156250,   2865625,   1478125,    573125,    166250,    35525,     5425,     560,     35,      1;
  8:  390625,  3125000, 11562500,  26250000,  40906250,  46375000,  39550000,  25875000,  13121875,   5175000,  1582000,   371000,   65450,   8400,    740,    40,   1;
  9: 1953125, 17578125, 73828125, 192187500, 347343750, 462656250, 470531250, 373500000, 234534375, 117359375, 46906875, 14940000, 3764250, 740250, 111150, 12300, 945, 45, 1;

The row polynomials satisfy P(n,x) = (5+5*x+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 + 5*x + 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 + 5*x + 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-65aaf6e4286453d2
• 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-65aaf6e4286453d2, e-da85173d9f94ea4d
• 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-65aaf6e4286453d2, e-da85173d9f94ea4d
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-da85173d9f94ea4d

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-65aaf6e4286453d2

  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)
• preserver_image affine, shift 0 e-da85173d9f94ea4d

  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]
    • ↵ preserver_image reversal, shift 0 e-0e058c428dfdfeb6 (expanded above)
    • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

• A272866 → x ↦ x/(x+1)