A081580 → x ↦ x/(x+1)

OEIS matches ?

• A015528: 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;
  1: 1,  2;
  2: 1,  9,    9;
  3: 1, 16,   42,    28;
  4: 1, 23,  124,   202,   101;
  5: 1, 30,  255,   740,   855,    342;
  6: 1, 37,  435,  1985,  3965,   3567,   1189;
  7: 1, 44,  664,  4280, 12910,  19472,  14308,   4088;
  8: 1, 51,  942,  7968, 33570,  75042,  90912,  56484,  14121;
  9: 1, 58, 1269, 13392, 74226, 228132, 402906, 407208, 219069, 48682;

The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (7*n^2-18*n+12+(52*n^2-155*n+108)*x+(162*n^2-513*n+360)*x^2+(268*n^2-861*n+600)*x^3+(247*n^2-785*n+540)*x^4+(120*n^2-372*n+252)*x^5+(24*n^2-72*n+48)*x^6)*P(n-1,x) - (n-1+(30*n-65)*x+(182*n-406)*x^2+(496*n-1090)*x^3+(729*n-1565)*x^4+(602*n-1261)*x^5+(264*n-540)*x^6+(48*n-96)*x^7)*d/dx P(n-1,x) + (x+29*x^2+154*x^3+370*x^4+485*x^5+361*x^6+144*x^7+24*x^8)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*D[P[n - 1, x], x] + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((7*n^2 - 18*n + 12 + (52*n^2 - 155*n + 108)*x + (162*n^2 - 513*n + 360)*x^2 + (268*n^2 - 861*n + 600)*x^3 + (247*n^2 - 785*n + 540)*x^4 + (120*n^2 - 372*n + 252)*x^5 + (24*n^2 - 72*n + 48)*x^6)*v[i - 1] - (n - 1 + (30*n - 65)*x + (182*n - 406)*x^2 + (496*n - 1090)*x^3 + (729*n - 1565)*x^4 + (602*n - 1261)*x^5 + (264*n - 540)*x^6 + (48*n - 96)*x^7)*deriv(v[i - 1], x) + (x + 29*x^2 + 154*x^3 + 370*x^4 + 485*x^5 + 361*x^6 + 144*x^7 + 24*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*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-28cdd807988aaaf4
• 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-28cdd807988aaaf4
• 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-28cdd807988aaaf4

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-28cdd807988aaaf4

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Pascal-(1,5,1) array.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang_derivative_lag e-307703197d4a988b root: theorem check

      Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

    • preserver_image reversal, shift 0 e-bf7624d7e60440c3

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Pascal-(1,5,1) array.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang_derivative_lag e-307703197d4a988b (expanded above)

Related sequences ?

Same fingerprint (1)

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

• A081580 → basis A007318

References

1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003