basis-map coefficients of A026729 → basis A056242 → (θ+1)²

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:
   1: 1;
   2: 1,   8;
   3: 1,  20,   36;
   4: 1,  36,  144,   128;
   5: 1,  56,  369,   704,    400;
   6: 1,  80,  765,  2336,   2800,   1152;
   7: 1, 108, 1395,  6032,  11400,   9792,   3136;
   8: 1, 140, 2331, 13328,  35200,  47232,  31360,   8192;
   9: 1, 176, 3654, 26432,  91225, 169632, 174832,  94208,  20736;
  10: 1, 216, 5454, 48384, 209025, 504072, 715792, 595968, 269568, 51200;

The row polynomials satisfy (36*n-36-(22*n^3-108*n^2+122*n-36)*x+(n^5-11*n^4+35*n^3-73*n^2+96*n-48)*x^2+(4*n^4-28*n^3+44*n^2-20*n)*x^3)*P(n,x) = (36*n-36-(22*n^3-108*n^2+122*n-324)*x+(n^5-11*n^4+35*n^3-225*n^2+760*n-384)*x^2+(10*n^4-96*n^3+238*n^2-280*n+96)*x^3+(16*n^3-80*n^2)*x^4)*P(n-1,x) + (72*x-(44*n^2-172*n+48)*x^2+(2*n^4-20*n^3+14*n^2+36*n-72)*x^3+(2*n^4-12*n^3+18*n^2-136*n+48)*x^4+(8*n^3-40*n^2)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 324)*x + (n^5 - 11*n^4 + 35*n^3 - 225*n^2 + 760*n - 384)*x^2 + (10*n^4 - 96*n^3 + 238*n^2 - 280*n + 96)*x^3 + (16*n^3 - 80*n^2)*x^4)*P[n - 1, x] + (72*x - (44*n^2 - 172*n + 48)*x^2 + (2*n^4 - 20*n^3 + 14*n^2 + 36*n - 72)*x^3 + (2*n^4 - 12*n^3 + 18*n^2 - 136*n + 48)*x^4 + (8*n^3 - 40*n^2)*x^5)*D[P[n - 1, x], x])/(36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 36)*x + (n^5 - 11*n^4 + 35*n^3 - 73*n^2 + 96*n - 48)*x^2 + (4*n^4 - 28*n^3 + 44*n^2 - 20*n)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 324)*x + (n^5 - 11*n^4 + 35*n^3 - 225*n^2 + 760*n - 384)*x^2 + (10*n^4 - 96*n^3 + 238*n^2 - 280*n + 96)*x^3 + (16*n^3 - 80*n^2)*x^4)*v[i - 1] + (72*x - (44*n^2 - 172*n + 48)*x^2 + (2*n^4 - 20*n^3 + 14*n^2 + 36*n - 72)*x^3 + (2*n^4 - 12*n^3 + 18*n^2 - 136*n + 48)*x^4 + (8*n^3 - 40*n^2)*x^5)*deriv(v[i - 1], x))/(36*n - 36 - (22*n^3 - 108*n^2 + 122*n - 36)*x + (n^5 - 11*n^4 + 35*n^3 - 73*n^2 + 96*n - 48)*x^2 + (4*n^4 - 28*n^3 + 44*n^2 - 20*n)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-6c5bdfbf185d1b6a
• Coefficients of one sign. Every nonzero row, n >= 1: 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-6c5bdfbf185d1b6a
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-6c5bdfbf185d1b6a

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 euler_polynomial, shift 0 e-6c5bdfbf185d1b6a

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A056242 e-3cb5f3d64c4bb317

      Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in [0, 0]
        no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.