A136572 → (θ+1)²

OEIS matches ?

• A052633: row sums, 13 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001563: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A094304: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A094258: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A052655: row sums, 13 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: 0, 4;
  2: 0, 0, 18;
  3: 0, 0,  0, 96;
  4: 0, 0,  0,  0, 600;
  5: 0, 0,  0,  0,   0, 4320;
  6: 0, 0,  0,  0,   0,    0, 35280;
  7: 0, 0,  0,  0,   0,    0,     0, 322560;
  8: 0, 0,  0,  0,   0,    0,     0,      0, 3265920;
  9: 0, 0,  0,  0,   0,    0,     0,      0,       0, 36288000;

The row polynomials satisfy n*P(n,x) = (n^2+2*n+1)*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[((n^2 + 2*n + 1)*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] = ((n^2 + 2*n + 1)*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-76872140ee232c50
• 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-76872140ee232c50
• 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-76872140ee232c50

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-76872140ee232c50

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

  • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers root_domain e-8128a83cc2062067 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-7d7235cf57d80f69

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: row n consists of n zeros followed by n!.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers root_domain e-8128a83cc2062067 (expanded above)

Related sequences ?

Same fingerprint (75)

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

• A010054 → reverse → θ+1
• A135838 → γ
• A010054 → reverse → θ+1
• A010054 → reverse → D
• A010054 → reverse → θ+1
• A136572 → θ+1
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• basis-map coefficients of A021012 → θ+1
• A085881 → γ
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• basis-map coefficients of A112227 → (θ+1)²
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A023531 → (θ+1)² → reverse
• basis-map coefficients of A030528, shifted → (θ+1)²
• A356546 → γ
• A010054 → reverse → D
• basis-map coefficients of A026729, shifted → (θ+1)²
• A038325 → γ
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A178821 → γ
• basis-map coefficients of A112227 → θ+1
• basis-map coefficients of A112227 → D
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• basis-map coefficients of A026729 → (θ+1)²
• A133080 → Veronese 2
• A178822 → γ
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• A038221 → γ
• basis-map coefficients of A297191, shifted → θ+1
• basis-map coefficients of A297191, shifted → (θ+1)²
• A010054 → reverse → (θ+1)²
• basis-map coefficients of A021012 → reverse
• basis-map coefficients of A030528, shifted → θ+1
• A010054 → reverse → (θ+1)²
• A023531 → (θ+1)² → (θ+1)²
• A344565 → γ
• A023531 → θ+1
• A023531 → (θ+1)² → θ+1
• A010054 → reverse → reverse
• basis-map coefficients of A026729, shifted → D
• basis-map coefficients of A021012 → D
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A130123 → reverse
• A007318 → D → γ
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• A010054 → reverse → D
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A131431: 3n + 1 preceded by n zeros.
• A130123 → D
• basis-map coefficients of A021012 → (θ+1)²
• A196347 → γ
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• basis-map coefficients of A297191, shifted → D
• A136572 → D
• A023531 → D → (θ+1)²
• A028326 → γ
• basis-map coefficients of A026729, shifted → reverse
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• A023531 → (θ+1)² → D
• A130123 → θ+1
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• A023531 → 2D+1+2n → Veronese 2 (j=1)
• A178820 → γ
• basis-map coefficients of A026729 → D
• A130123 → (θ+1)²
• basis-map coefficients of A026729, shifted → θ+1
• basis-map coefficients of A030528, shifted → D
• A023531 → 2D+1+2n → Veronese 2
• A010054 → reverse → reverse
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• A156992 → γ
• basis-map coefficients of A scaled Hermite triangle.
• A038338 → γ
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A030528, shifted → reverse

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.