A088996 → basis A007318

OEIS matches ?

• A008544: 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,         1;
  2:        3,         5,         2;
  3:       15,        34,        25,         6;
  4:      105,       298,       311,       142,        24;
  5:      945,      3207,      4289,      2833,       926,       120;
  6:    10395,     40947,     66421,     56897,     27184,      6876,       720;
  7:   135135,    605076,   1150102,   1204608,    751671,    279676,     57492,     5040;
  8:  2027025,  10157220,  22092138,  27269936,  20911929,  10208508,   3099788,   535536,   40320;
  9: 34459425, 190915965, 466981326, 662418154, 600932217, 361751997, 144572968, 37002204, 5505264, 362880;

The row polynomials satisfy P(n,x) = (2*n-1+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[(2*n - 1 + 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] = (2*n - 1 + 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-5ca25d3c8ee8ed44
• 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-5ca25d3c8ee8ed44
• 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-5ca25d3c8ee8ed44

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 A007318 e-5ca25d3c8ee8ed44

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-4c0875826685c298 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-e5c0b6618046e013

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers root_domain e-edbf92ee2e446027 root: theorem check

          Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A109692 → basis A013609