A202840 -id:A202840 - cp's OEIS frontend

A202838 Triangle read by rows: T(n,k) is the number of secondary structures of size n (n>=0) having k stacks of length 1 (k>=0).

1, 1, 1, 1, 1, 1, 3, 2, 6, 4, 10, 3, 8, 15, 14, 14, 27, 40, 1, 23, 56, 90, 16, 38, 122, 178, 85, 65, 253, 356, 295, 9, 117, 494, 762, 805, 105, 214, 938, 1713, 1912, 594, 2, 391, 1783, 3828, 4326, 2331, 76, 708, 3456, 8265, 9882, 7290, 771, 1278, 6793, 17309, 23109, 19784, 4529, 30

Offset: 0

Emeric Deutsch, Dec 25 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Sum of entries in row n is A004148 (the secondary structure numbers).
Sum(k*T(n,k), k>=0)=A202839(n).
T(n,0)=A202840(n).

			Row 5 is 2,6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.
Triangle starts:
1;
1;
1;
1,1;
1,3;
2,6;
4,10,3;
8,15,14;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A202839, A202840, A202841, A202842, A202843, A202844

f := (t-1)*z^2+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f. G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^2 + z^2/(1-z^2).

The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A202839 Number of stacks of length 1 in all 2ndary structures of size n.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 16, 43, 110, 284, 733, 1886, 4853, 12486, 32121, 82647, 212699, 547552, 1410023, 3632260, 9360140, 24129284, 62224692, 160522287, 414246823, 1069376386, 2761502201, 7133442743, 18432633823, 47643696626, 123182434292, 318575889057, 824125660356

Offset: 0

Emeric Deutsch, Dec 25 2011

nonn

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Number of stacks of length 2 in all 2ndary structures of size n+2.
Number of stacks of length 3 in all 2ndary structures of size n+4.

			a(5)=6: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of length 1, respectively.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A202838, A202840, A202841, A202842, A202843, A202844

g := z^2*(1-z^2)^2*S*(S-1)/(1-z+z^2-2*z^2*S): S := ((1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4))*1/2)/z^2: gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32);

CoefficientList[Series[-(1 - x^2)^2 * ((1 - x) + (-1 + 2*x + x^3) / Sqrt[(1 - 3*x + x^2) * (1 + x + x^2)]) / (2*x^2), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)

a(n) = Sum(k*A202838(n,k), k>=0).
a(n) = Sum(k*A202841(n+2,k), k>=0).
a(n) = Sum(k*A202843(n+4,k), k>=0).
G.f.: g(z) = z^2*(1-z^2)^2*S(S - 1)/(1 - z + z^2 -2*z^2*S), where S is defined by S = 1 + z*S + z^2*S(S-1) (the g.f. of the secondary structure numbers A004148).
a(n) ~ 5^(3/4) * phi^(2*n-3) / (2*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022
D-finite with recurrence -(n+2)*(406*n-3981)*a(n) +(2022*n^2-15917*n-13552)*a(n-1) +4*(-402*n^2+2594*n+593)*a(n-2) +4*(-605*n^2+7719*n-23415)*a(n-3) +4*(-203*n^2-527*n+15295)*a(n-4) +2*(804*n^2-8404*n+14555)*a(n-5) +(2826*n^2-42913*n+153174)*a(n-6) -(1210*n-6753)*(n-10)*a(n-7)=0. - R. J. Mathar, Jul 26 2022

A202841 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 2 (n>=0, k>=0).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 142, 43, 316, 104, 3, 708, 256, 14, 1593, 647, 43, 3625, 1610, 138, 8314, 3990, 430, 1, 19165, 9944, 1247, 16, 44433, 24762, 3552, 85, 103557, 61574, 10040, 331, 242376, 153270, 27877, 1225, 569514, 381718, 76491, 4272, 9

Offset: 0

Emeric Deutsch, Dec 25 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

Sum of entries in row n is A004148 (the secondary structure numbers).

			Row 5 is 7,1: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; only the last one has a stack of length 2.
Triangle starts:
1;
1;
1;
2;
4;
7,1;
14,3;
31,6;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004148, A202838, A202839, A202840, A202842, A202843, A202844

f := (t-1)*z^4+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

Sum(k*T(n,k), k>=0) = A202839(n-2).
T(n,0) = A202842(n).
G.f. G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^4 + z^2/(1-z^2).
The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + [f/(1 + f)]H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A202842 Number of secondary structures of size n having no stacks of length 2.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 31, 66, 142, 316, 708, 1593, 3625, 8314, 19165, 44433, 103557, 242376, 569514, 1343099, 3177766, 7540845, 17943506, 42804078, 102345017, 245233366, 588785677, 1416247791, 3412495415, 8235829927, 19906780104, 48185131721, 116790380824, 283432579807

Offset: 0

Emeric Deutsch, Dec 25 2011

nonn

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

a(n) = A202841(n,0).

			a(5)=7; representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; only the last one has a stack of length 2.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A202838, A202839, A202840, A202841, A202843, A202844

f := z^2*(1-z^2+z^4)/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 38)): seq(coeff(Gser, z, n), n = 0 .. 34);

G.f.: G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2*(1-z^2+z^4)/(1-z^2).

A202843 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 1, 79, 3, 179, 6, 407, 16, 935, 43, 2173, 110, 5089, 284, 12005, 727, 3, 28500, 1858, 14, 68022, 4767, 43, 163154, 12210, 138, 393060, 31255, 433, 950652, 80057, 1295, 2307454, 205088, 3804, 1, 5618906, 525534, 10985, 16, 13723145, 1347174, 31297, 85

Offset: 0

Emeric Deutsch, Dec 25 2011

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
Sum of entries in row n is A004148 (the secondary structure numbers).
Sum(k*T(n,k), k>=0) = A202839(n-4).
T(n,0) = A202844(n).

			Row 5 is 8: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; none of them has stacks of length 3.
Triangle starts:
1;
1;
1;
2;
4;
8;
17;
36,1;
79,3;

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A004148, A202838, A202839, A202840, A202841, A202842, A202844

f := (t-1)*z^6+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 26)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 22 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form

G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^6 + z^2/(1-z^2).

The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + [f/(1 + f)]H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .

A202844 Number of secondary structures of size n having no stacks of length 3.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 79, 179, 407, 935, 2173, 5089, 12005, 28500, 68022, 163154, 393060, 950652, 2307454, 5618906, 13723145, 33607242, 82507764, 203028034, 500659653, 1237053269, 3062204227, 7593229687, 18858944533, 46909741893, 116848688876, 291449697298

Offset: 0

Emeric Deutsch, Dec 25 2011

nonn

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

			a(5)=8; representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; none of them has stacks of length 3.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A202838, A202839, A202840, A202841, A202842, A202843.

f := z^2*(1-z^4+z^6)/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 37)): seq(coeff(Gser, z, n), n = 0 .. 33);

G.f.: G=G(z) satisfies G = 1+zG +fG(G-1)/(1+f), where f = z^2*(1-z^4+z^6)/(1-z^2).

a(n) = A202843(n,0).

cp's OEIS Frontend

A202838 Triangle read by rows: T(n,k) is the number of secondary structures of size n (n>=0) having k stacks of length 1 (k>=0).

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A202839 Number of stacks of length 1 in all 2ndary structures of size n.

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A202841 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 2 (n>=0, k>=0).

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A202842 Number of secondary structures of size n having no stacks of length 2.

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A202843 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 3.

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A202844 Number of secondary structures of size n having no stacks of length 3.

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