A172431 → ⊙ C(n,k+1) (shift -1) → reverse

OEIS matches ?

• A106228: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A391045: reversed triangle, 66 terms agree, 11 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 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:
   1: 1;
   2: 1,   1;
   3: 1,   4,    1;
   4: 1,  10,    9,     1;
   5: 1,  20,   42,    16,     1;
   6: 1,  35,  140,   120,    25,     1;
   7: 1,  56,  378,   600,   275,    36,     1;
   8: 1,  84,  882,  2310,  1925,   546,    49,    1;
   9: 1, 120, 1848,  7392, 10010,  5096,   980,   64,  1;
  10: 1, 165, 3564, 20592, 42042, 34398, 11760, 1632, 81, 1;

The row polynomials satisfy (4*n^3-2*n^2-2*n)*P(n,x) = (4*n^3-2*n^2-2*n+(7*n^3-11*n^2+4*n)*x)*P(n-1,x) + ((12*n^2+18*n-12)*x+(2*n^2-14*n+6)*x^2)*d/dx P(n-1,x) + ((20*n-8)*x^2-(5*n-2)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 2*n^2 - 2*n + (7*n^3 - 11*n^2 + 4*n)*x)*P[n - 1, x] + ((12*n^2 + 18*n - 12)*x + (2*n^2 - 14*n + 6)*x^2)*D[P[n - 1, x], x] + ((20*n - 8)*x^2 - (5*n - 2)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 2*n^2 - 2*n)]];
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] = ((4*n^3 - 2*n^2 - 2*n + (7*n^3 - 11*n^2 + 4*n)*x)*v[i - 1] + ((12*n^2 + 18*n - 12)*x + (2*n^2 - 14*n + 6)*x^2)*deriv(v[i - 1], x) + ((20*n - 8)*x^2 - (5*n - 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n)); 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)

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,    1;
  3: 1,  9,   10,     1;
  4: 1, 16,   42,    20,     1;
  5: 1, 25,  120,   140,    35,     1;
  6: 1, 36,  275,   600,   378,    56,     1;
  7: 1, 49,  546,  1925,  2310,   882,    84,    1;
  8: 1, 64,  980,  5096, 10010,  7392,  1848,  120,   1;
  9: 1, 81, 1632, 11760, 34398, 42042, 20592, 3564, 165, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where (4*n^3+10*n^2+6*n)*P(n,x) = (4*n^3+10*n^2+6*n+(7*n^3+10*n^2+3*n)*x)*P(n-1,x) + ((12*n^2+42*n+18)*x+(2*n^2-10*n-6)*x^2)*d/dx P(n-1,x) + ((20*n+12)*x^2-(5*n+3)*x^3)*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[((4*n^3 + 10*n^2 + 6*n + (7*n^3 + 10*n^2 + 3*n)*x)*P[n - 1, x] + ((12*n^2 + 42*n + 18)*x + (2*n^2 - 10*n - 6)*x^2)*D[P[n - 1, x], x] + ((20*n + 12)*x^2 - (5*n + 3)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^3 + 10*n^2 + 6*n)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^3 + 10*n^2 + 6*n + (7*n^3 + 10*n^2 + 3*n)*x)*v[i - 1] + ((12*n^2 + 42*n + 18)*x + (2*n^2 - 10*n - 6)*x^2)*deriv(v[i - 1], x) + ((20*n + 12)*x^2 - (5*n + 3)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^3 + 10*n^2 + 6*n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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 >= 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-d1e8a832158fe845
• 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-d1e8a832158fe845
• 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-d1e8a832158fe845

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 reversal, shift 0 e-d1e8a832158fe845

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image hadamard_multiplier, shift -1 e-1491609a6fc5f8c9

      Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

      • premise (sequence): Even row Pascal-square read by antidiagonals.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-83282836b5ac538e root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (1)

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

• A391045: Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023