A272866 → basis A013611

OEIS matches ?

• A009985: 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,       20,         16;
  2:      25,      200,        560,         640,         256;
  3:     125,     1500,       7200,       17600,       23040,        15360,          4096;
  4:     625,    10000,      68000,      256000,      582400,       819200,        696320,        327680,          65536;
  5:    3125,    62500,     550000,     2800000,     9120000,     19840000,      29184000,      28672000,       18022400,        6553600,        1048576;
  6:   15625,   375000,    4050000,    26000000,   110400000,    326400000,     688640000,    1044480000,     1130496000,      851968000,      424673280,      125829120,       16777216;
  7:   78125,  2187500,   28000000,   217000000,  1136800000,   4256000000,   11737600000,   24217600000,    37560320000,    43581440000,    37250662400,    22754099200,     9395240960,     2348810240,      268435456;
  8:  390625, 12500000,  185000000,  1680000000, 10472000000,  47488000000,  161996800000,  423936000000,   859955200000,  1356595200000,  1658847232000,  1556086784000,  1098068787200,   563714457600,   198642237440,    42949672960,    4294967296;
  9: 1953125, 70312500, 1181250000, 12300000000, 88920000000, 473760000000, 1927296000000, 6119424000000, 15370444800000, 30765056000000, 49185423360000, 62662901760000, 63153635328000, 49677336576000, 29836600934400, 13207024435200, 4058744094720, 773094113280, 68719476736;

The row polynomials satisfy P(n,x) = (5+20*x+16*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 + 20*x + 16*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 + 20*x + 16*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-a99971f11d0aad5a
• 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-a99971f11d0aad5a
• 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-a99971f11d0aad5a

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 A013611 e-a99971f11d0aad5a

  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.