A272866 → basis A013609

OEIS matches ?

• A001029: 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,       10,         4;
  2:      25,      100,       140,         80,         16;
  3:     125,      750,      1800,       2200,       1440,         480,          64;
  4:     625,     5000,     17000,      32000,      36400,       25600,       10880,        2560,         256;
  5:    3125,    31250,    137500,     350000,     570000,      620000,      456000,      224000,       70400,       12800,        1024;
  6:   15625,   187500,   1012500,    3250000,    6900000,    10200000,    10760000,     8160000,     4416000,     1664000,      414720,       61440,        4096;
  7:   78125,  1093750,   7000000,   27125000,   71050000,   133000000,   183400000,   189200000,   146720000,    85120000,    36377600,    11110400,     2293760,     286720,      16384;
  8:  390625,  6250000,  46250000,  210000000,  654500000,  1484000000,  2531200000,  3312000000,  3359200000,  2649600000,  1619968000,   759808000,   268083200,   68812800,   12124160,   1310720,    65536;
  9: 1953125, 35156250, 295312500, 1537500000, 5557500000, 14805000000, 30114000000, 47808000000, 60040800000, 60088000000, 48032640000, 30597120000, 15418368000, 6064128000, 1821081600, 403046400, 61931520, 5898240, 262144;

The row polynomials satisfy P(n,x) = (5+10*x+4*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 + 10*x + 4*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 + 10*x + 4*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-89275777750acd05
• 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-89275777750acd05
• 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-89275777750acd05

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 A013609 e-89275777750acd05

  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.