A136572 → basis A013611

OEIS matches ?

• A052562: 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;
  2:      2,       16,        32;
  3:      6,       72,       288,        384;
  4:     24,      384,      2304,       6144,        6144;
  5:    120,     2400,     19200,      76800,      153600,      122880;
  6:    720,    17280,    172800,     921600,     2764800,     4423680,      2949120;
  7:   5040,   141120,   1693440,   11289600,    45158400,   108380160,    144506880,     82575360;
  8:  40320,  1290240,  18063360,  144506880,   722534400,  2312110080,   4624220160,   5284823040,   2642411520;
  9: 362880, 13063680, 209018880, 1950842880, 11705057280, 46820229120, 124853944320, 214035333120, 214035333120, 95126814720;

The row polynomials satisfy P(n,x) = (n+4*n*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 4*n*x)*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] = (n + 4*n*x)*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-d759043f254c4b9e
• 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-d759043f254c4b9e
• 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-d759043f254c4b9e

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-d759043f254c4b9e

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-8128a83cc2062067 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-7d7235cf57d80f69

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)

Related sequences ?

Same fingerprint (14)

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

• basis-map coefficients of A030528, shifted → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• A038330 → basis A007318
• basis-map coefficients of A112227 → basis A013611
• A027466 → basis A013609
• A013611 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• basis-map coefficients of A026729 → basis A013611
• A027467 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A130123 → basis A013611
• basis-map coefficients of A021012 → basis A013611
• A038231 → D