A064861 → x ↦ x/(x+1)

OEIS matches ?

• A282023: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,  5,   6;
  1: 1,  8,  21,   18;
  2: 1, 10,  37,   60,    36;
  3: 1, 13,  67,  171,   216,   108;
  4: 1, 15,  93,  305,   558,   540,    216;
  5: 1, 18, 138,  584,  1473,  2214,   1836,    648;
  6: 1, 20, 174,  860,  2641,  5160,   6264,   4320,   1296;
  7: 1, 23, 234, 1382,  5221, 13083,  21744,  23112,  14256,   3888;
  8: 1, 25, 280, 1850,  7985, 23525,  47910,  66600,  60480,  32400,   7776;
  9: 1, 28, 355, 2690, 13535, 47480, 118485, 210330, 260280, 213840, 104976, 23328;

The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+(5*n+11)*x+(27*n+39)*x^2+(51*n+61)*x^3+(41*n+44)*x^4+(12*n+12)*x^5)*P(n-1,x) - (2*x+16*x^2+48*x^3+68*x^4+46*x^5+12*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 5*x + 6*x^2.

P[0, x_] := 1 + 5*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (5*n + 11)*x + (27*n + 39)*x^2 + (51*n + 61)*x^3 + (41*n + 44)*x^4 + (12*n + 12)*x^5)*P[n - 1, x] - (2*x + 16*x^2 + 48*x^3 + 68*x^4 + 46*x^5 + 12*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 5*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (5*n + 11)*x + (27*n + 39)*x^2 + (51*n + 61)*x^3 + (41*n + 44)*x^4 + (12*n + 12)*x^5)*v[i - 1] - (2*x + 16*x^2 + 48*x^3 + 68*x^4 + 46*x^5 + 12*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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 [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-5c2afa55aadc0623
• 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-5c2afa55aadc0623
• 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-5c2afa55aadc0623

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

• preserver_image mobius, shift 0 e-5c2afa55aadc0623

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-bbffc3f85cdc52e2 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.

• A152842 → basis A013611