A001263 → θ+1

OEIS matches ?

• A132813: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · view at the OEIS · generated b-file 990 terms from n = 0 ?
• A088218: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A001700: 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 ?
• A103371: reversed triangle, 60 terms agree, 10 full rows row-aligned (tabl) OEIS nonn · 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:
  0: 1;
  1: 1,  2;
  2: 1,  6,    3;
  3: 1, 12,   18,     4;
  4: 1, 20,   60,    40,     5;
  5: 1, 30,  150,   200,    75,     6;
  6: 1, 42,  315,   700,   525,   126,     7;
  7: 1, 56,  588,  1960,  2450,  1176,   196,    8;
  8: 1, 72, 1008,  4704,  8820,  7056,  2352,  288,   9;
  9: 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10;

The row polynomials satisfy (n^2+n)*P(n,x) = (n^2+n+(3*n^2+n)*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx 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 + n + (3*n^2 + n)*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + 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 + n + (3*n^2 + n)*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + 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)

Triangle begins:
  0:  1;
  1:  2,   1;
  2:  3,   6,    1;
  3:  4,  18,   12,     1;
  4:  5,  40,   60,    20,     1;
  5:  6,  75,  200,   150,    30,     1;
  6:  7, 126,  525,   700,   315,    42,     1;
  7:  8, 196, 1176,  2450,  1960,   588,    56,    1;
  8:  9, 288, 2352,  7056,  8820,  4704,  1008,   72,  1;
  9: 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n^2+n)*P(n,x) = (n^2+n+(3*n^2+n)*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx 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 + n + (3*n^2 + n)*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + 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] = ((n^2 + n + (3*n^2 + n)*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + 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)

Triangle begins:
  0: 1;
  1: 1,  2;
  2: 1,  6,    3;
  3: 1, 12,   18,     4;
  4: 1, 20,   60,    40,     5;
  5: 1, 30,  150,   200,    75,     6;
  6: 1, 42,  315,   700,   525,   126,     7;
  7: 1, 56,  588,  1960,  2450,  1176,   196,    8;
  8: 1, 72, 1008,  4704,  8820,  7056,  2352,  288,   9;
  9: 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10;

The row polynomials satisfy (n^2+n)*P(n,x) = (n^2+n+(3*n^2+n)*x)*P(n-1,x) + ((2*n+1)*x-(2*n+1)*x^2)*d/dx 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 + n + (3*n^2 + n)*x)*P[n - 1, x] + ((2*n + 1)*x - (2*n + 1)*x^2)*D[P[n - 1, x], x])/(n^2 + n)]];
Table[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] = ((n^2 + n + (3*n^2 + n)*x)*v[i - 1] + ((2*n + 1)*x - (2*n + 1)*x^2)*deriv(v[i - 1], x))/(n^2 + n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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 >= 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-d2a71b42a97e9c64
• 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-d2a71b42a97e9c64
• 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-d2a71b42a97e9c64
• 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-d2a71b42a97e9c64

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-d2a71b42a97e9c64

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

  • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-d285019b29e04549 root: theorem check

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

    • preserver_image reversal, shift 0 e-14da04f2a2dce119

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan tri…
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check three_term_recurrence e-d285019b29e04549 (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A367270: Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
• A008459 → D
• narayana → θ+1

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