A356546 → basis A007318

OEIS matches ?

• A098658: 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:        4,         2;
  2:       24,        24,         6;
  3:      160,       240,       120,        20;
  4:     1120,      2240,      1680,       560,        70;
  5:     8064,     20160,     20160,     10080,      2520,      252;
  6:    59136,    177408,    221760,    147840,     55440,    11088,      924;
  7:   439296,   1537536,   2306304,   1921920,    960960,   288288,    48048,    3432;
  8:  3294720,  13178880,  23063040,  23063040,  14414400,  5765760,  1441440,  205920,  12870;
  9: 24893440, 112020480, 224040960, 261381120, 196035840, 98017920, 32672640, 7001280, 875160, 48620;

The row polynomials satisfy n*P(n,x) = (8*n-4+(4*n-2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 4 + (4*n - 2)*x)*P[n - 1, x])/(n)]];
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] = ((8*n - 4 + (4*n - 2)*x)*v[i - 1])/(n)); 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-ced3e67c3d44f436
• 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-ced3e67c3d44f436
• 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-ced3e67c3d44f436
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-ced3e67c3d44f436

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-ced3e67c3d44f436

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-883c9c594ff729de root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Related sequences ?

Same fingerprint (67)

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

• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A007318 → D → x ↦ x/(x+1)
• A085880 → x ↦ x/(x+1)
• A038208 → basis A007318
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A027466 → basis A013611
• basis-map coefficients of A113214, shifted → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A156992 → basis A007318
• A136572 → basis A013609
• A159854 → x ↦ x/(x+1)
• A135838 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A038325 → basis A013611
• A013609 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A120910 → D
• A038338 → basis A007318
• A135838 → x ↦ x/(x+1)
• A344565 → x ↦ x/(x+1)
• A178822 → basis A013611
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A038243 → basis A013610
• A028326 → x ↦ x/(x+1)
• A178822 → x ↦ x/(x+1)
• A085880 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A130123 → basis A013609
• A196347 → basis A013611
• A013609 → D
• A038325 → basis A007318
• A135838 → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013609
• basis-map coefficients of A021012 → basis A013609
• A085881 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609
• A344565 → basis A013611
• A178821 → basis A013611
• A178820 → basis A013611
• A120909 → basis A013610
• A085881 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609
• A178820 → basis A007318
• A178822 → x ↦ x+1
• A038207 → D
• basis-map coefficients of A026729 → basis A013609
• A120909 → D
• A156992 → basis A013611
• A085880 → x ↦ x+1
• A038338 → basis A013611
• A196347 → basis A007318
• A038330 → basis A013609
• A038208 → x ↦ x/(x+1)
• A038208 → basis A013611
• A178821 → basis A007318
• A038221 → basis A013611
• A028326 → basis A013611
• A178821 → x ↦ x/(x+1)
• A028326 → basis A007318
• A344565 → basis A007318
• A356546 → basis A013611
• A178820 → x ↦ x/(x+1)

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