A178821 → basis A007318

OEIS matches ?

• A036217: 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:     10,       5;
  2:     60,      60,      15;
  3:    280,     420,     210,      35;
  4:   1120,    2240,    1680,     560,      70;
  5:   4032,   10080,   10080,    5040,    1260,     126;
  6:  13440,   40320,   50400,   33600,   12600,    2520,    210;
  7:  42240,  147840,  221760,  184800,   92400,   27720,   4620,    330;
  8: 126720,  506880,  887040,  887040,  554400,  221760,  55440,   7920,   495;
  9: 366080, 1647360, 3294720, 3843840, 2882880, 1441440, 480480, 102960, 12870, 715;

The row polynomials satisfy n*P(n,x) = (2*n+8+(n+4)*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[((2*n + 8 + (n + 4)*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] = ((2*n + 8 + (n + 4)*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-484d449f4d0f99c5
• 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-484d449f4d0f99c5, e-a119db10a83a7e8e
• 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-484d449f4d0f99c5, e-a119db10a83a7e8e
• 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-484d449f4d0f99c5, e-a119db10a83a7e8e
• 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-a119db10a83a7e8e

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-484d449f4d0f99c5

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-20c79d8930f25d5b root: theorem check

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

    • preserver_image reversal, shift 0 e-12f9ed6b8198f9ce

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-20c79d8930f25d5b (expanded above)
• preserver_image affine, shift 0 e-a119db10a83a7e8e

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ theorem_check liu_wang e-20c79d8930f25d5b (expanded above)
    • ↵ preserver_image reversal, shift 0 e-12f9ed6b8198f9ce (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A120909 → basis A013610
• A007318 → D → x ↦ x/(x+1)
• A038330 → basis A013609
• A038338 → basis A007318
• A159854 → x ↦ x/(x+1)
• A178821 → basis A013611
• A178822 → x ↦ x+1
• A028326 → basis A013611
• A085881 → basis A013611
• A038325 → basis A013611
• A038208 → basis A007318
• A178821 → x ↦ x/(x+1)
• A135838 → basis A013611
• A028326 → basis A007318
• A085881 → basis A007318
• 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…
• A178820 → basis A013611
• basis-map coefficients of A113214, shifted → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A028326 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013609
• 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-…
• basis-map coefficients of A026729 → basis A013609
• A196347 → basis A007318
• A038221 → basis A013611
• A085880 → basis A013611
• A178820 → x ↦ x/(x+1)
• A027466 → basis A013611
• A038338 → basis A013611
• A178822 → basis A013611
• A013609 → D
• A038208 → x ↦ x/(x+1)
• A130123 → basis A013609
• A344565 → x ↦ x/(x+1)
• A135838 → x ↦ x/(x+1)
• A038325 → basis A007318
• A085880 → x ↦ x+1
• A007318 → ×(1+2n+n²) → x ↦ x+1
• basis-map coefficients of A021012 → basis A013609
• A156992 → basis A013611
• A178820 → basis A007318
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A135838 → x ↦ x+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A120909 → D
• A356546 → basis A013611
• A178822 → x ↦ x/(x+1)
• A038207 → D
• A085880 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A196347 → basis A013611
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A038243 → basis A013610
• A038208 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609
• A344565 → basis A013611
• A156992 → basis A007318
• A344565 → basis A007318
• A120910 → D
• A356546 → basis A007318
• A136572 → basis A013609
• basis-map coefficients of A112227 → basis A013609
• A013609 → basis A013610
• A007318 → ×(1+(3/2)n+(1/2)n²) → 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