A026374 → x ↦ x+1

OEIS matches ?

No OEIS matches.

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:    2,    1;
  2:    5,    5,     1;
  3:   10,   15,     7,     1;
  4:   25,   50,    35,    10,    1;
  5:   50,  125,   120,    55,   12,    1;
  6:  125,  375,   450,   275,   90,   15,    1;
  7:  250,  875,  1275,  1000,  455,  120,   17,   1;
  8:  625, 2500,  4250,  4000, 2275,  800,  170,  20,  1;
  9: 1250, 5625, 11000, 12250, 8550, 3875, 1140, 210, 22, 1;

The row polynomials satisfy P(n,x) = (5+5*x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(5 + 5*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (5 + 5*x + x^2)*v[i - 2]); 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, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-aab7c9ef02515066
• 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-aab7c9ef02515066, e-d1e99a8e2ae807ea
• 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-aab7c9ef02515066, e-d1e99a8e2ae807ea
• 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-d1e99a8e2ae807ea

Proof chains

2 incoming proof steps. 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-d1e99a8e2ae807ea

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-6e3ca670df3d0a74 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-b2e3690a28625635

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)
• preserver_image affine, shift 0 e-aab7c9ef02515066

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)
    • ↵ preserver_image reversal, shift 0 e-b2e3690a28625635 (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A026374 → x ↦ x/(x+1)