basis-map coefficients of A297191, shifted → basis A013609

OEIS matches ?

• A100062: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A001019: 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: 1,  4,   4;
  2: 1,  8,  24,   32,    16;
  3: 1, 12,  60,  160,   240,    192,      64;
  4: 1, 16, 112,  448,  1120,   1792,    1792,    1024,      256;
  5: 1, 20, 180,  960,  3360,   8064,   13440,   15360,    11520,     5120,     1024;
  6: 1, 24, 264, 1760,  7920,  25344,   59136,  101376,   126720,   112640,    67584,    24576,     4096;
  7: 1, 28, 364, 2912, 16016,  64064,  192192,  439296,   768768,  1025024,  1025024,   745472,   372736,   114688,    16384;
  8: 1, 32, 480, 4480, 29120, 139776,  512512, 1464320,  3294720,  5857280,  8200192,  8945664,  7454720,  4587520,  1966080,   524288,    65536;
  9: 1, 36, 612, 6528, 48960, 274176, 1188096, 4073472, 11202048, 24893440, 44808192, 65175552, 76038144, 70189056, 50135040, 26738688, 10027008, 2359296, 262144;

The row polynomials satisfy P(n,x) = (1+4*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[(1 + 4*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] = (1 + 4*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-20c0b01c4fb345ed
• 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-20c0b01c4fb345ed
• 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-20c0b01c4fb345ed

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-20c0b01c4fb345ed

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)

Related sequences ?

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

• A034870 → x ↦ x/(x+1) (15 rows compared; no all-index equality certificate was found)

Same fingerprint (4)

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

• A139548 → basis A007318
• A034870 → basis A013611
• A034870 → x ↦ x+1
• A139548 → basis A013611