A030279 -id:A030279 - cp's OEIS frontend

A174654 Partial sums of A030279.

1, 9, 59, 335, 1732, 8404, 38969, 174637, 762063, 3255119, 13662512, 56511312, 230859729, 933104857, 3736739227, 14843364031, 58540883044, 229411923796, 893904838857, 3465221557357, 13370453635199

Offset: 1

Jonathan Vos Post, Mar 25 2010

Index entries for linear recurrences with constant coefficients, signature (12,-54,115,-132,108,-59,24,-6,1).

Cf. A030279.

LinearRecurrence[{12,-54,115,-132,108,-59,24,-6,1},{1,9,59,335,1732,8404,38969,174637,762063},30] (* Harvey P. Dale, Nov 07 2022 *)

a(n) = Sum_{i=1..n} A030279(i).

G.f.: x*(x-1)^2*(x+1)*(x^3-4*x^2+2*x-1) / (x^3-2*x^2+4*x-1)^3. - Colin Barker, Apr 21 2013

A030267 Compose the natural numbers with themselves, A(x) = B(B(x)) where B(x) = x/(1-x)^2 is the generating function for natural numbers.

Original entry on oeis.org

1, 4, 14, 46, 145, 444, 1331, 3926, 11434, 32960, 94211, 267384, 754309, 2116936, 5914310, 16458034, 45638101, 126159156, 347769719, 956238170, 2623278946, 7181512964, 19622668679, 53522804976, 145753273225, 396323283724, 1076167858046, 2918447861686

Offset: 1

Christian G. Bower

Sum of pyramid weights of all nondecreasing Dyck paths of semilength n. (A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.) Example: a(4) = 46. Indeed, there are 14 Dyck paths of semilength 4. One of them, namely UUDUDDUD is not nondecreasing because the valleys are at heights 1 and 0. The other 13, with the maximal pyramids shown between parentheses, are: (UD)(UD)(UD)(UD), (UD)(UD)(UUDD), (UD)(UUDD)(UD), (UD)U(UD)(UD)D, (UD)(UUUDDD), (UUDD)(UD)(UD), (UUDD)(UUDD), (UUUDDD)(UD), U(UD)(UD)(UD)D, U(UD)(UUDD)D, U(UUDD)(UD)D, UU(UD)(UD)DD and (UUUUDDDD). The pyramid weights of these paths are 4, 4, 4, 3, 4, 4, 4, 4, 3, 3, 3, 2, and 4, respectively. Their sum is 46. a(n) = Sum_{k = 1..n} k*A121462(n, k). - Emeric Deutsch, Jul 31 2006
Number of 1s in all compositions of n, where compositions are understood with two different kinds of 1s, say 1 and 1' (n >= 1). Example: a(2) = 4 because the compositions of 2 are 11, 11', 1'1, 1'1', 2, having a total of 2 + 1 + 1 + 0 + 0 = 4 1s. Also number of k's in all compositions of n + k (k = 2, 3, ...). - Emeric Deutsch, Jul 21 2008
From Petros Hadjicostas, Jun 24 2019: (Start)
If c = (c(m): m >= 1) is the input sequence and b_k = (b_k(n): n >= 1) is the output sequence under the AIK[k] = INVERT[k] transform (see Bower's web link below), then the bivariate g.f. of the list of sequences (b_k: k >= 1) = ((b_k(n): n >= 1): k >= 1) is Sum_{n, k >= 1} b_k(n)*x^n*y^k = y*C(x)/(1 - y*C(x)), where C(x) = Sum_{m >= 1} c(m)*x^m is the g.f. of the input sequence.
Here, b_k(n) is the number of all (linear) compositions of n with k parts where a part of size m is colored with one of c(m) colors. Thus, Sum_{k = 1..n} k*b_k(n) is the total number of parts in all compositions of n.
If we differentiate the bivariate g.f. function above, i.e., Sum_{n, k >= 1} b_k(n)*x^n*y^k, with respect to y and set y = 1, we get the g.f. of the sequence (Sum_{k = 1..n} k*b_k(n): n >= 1). It is C(x)/(1 - C(x))^2.
When c(m) = m for all m >= 1, we have m-color compositions of n that were first studied by Agarwal (2000). The cyclic version of these m-color compositions were studied by Gibson (2017) and Gibson et al. (2018).
When c(m) = m for each m >= 1, we have C(x) = x/(1 - x)^2, and so C(x)/(1 - C(x))^2 = x * (1 - x)^2/(1 - 3*x + x^2)^2, which is the g.f. of the current sequence.
Hence, a(n) is the total number of parts in all m-color compositions of n (in the sense of Agarwal (2000)).
(End)
Series reversal gives A153294 starting from index 1, with alternating signs: 1, -4, 18, -86, 427, -2180, ... - Vladimir Reshetnikov, Aug 03 2019

			From _Petros Hadjicostas_, Jun 24 2019: (Start)
Recall that with m-color compositions, a part of size m may be colored with one of m colors.
We have a(1) = 1 because we only have one colored composition, namely 1_1, that has only 1 part.
We have a(2) = 4 because we have the following colored compositions of n = 2: 2_1, 2_2, 1_1 + 1_1; hence, a(2) = 1 + 1 + 2 = 4.
We have a(3) = 14 because we have the following colored compositions of n = 3: 3_1, 3_2, 3_3, 1_1 + 2_1, 1_1 + 2_2, 2_1 + 1_1, 2_2 + 1_1, 1_1 + 1_1 + 1_1; hence, a(3) = 1 + 1 + 1 + 2 + 2 + 2 + 2 + 3 = 14.
We have a(14) = 46 because we have the following colored compositions of n = 4:
(i) 4_1, 4_2, 4_3, 4_4; with a total of 4 parts.
(ii) 1_1 + 3_1, 1_1 + 3_2, 1_1 + 3_3, 3_1 + 1_1, 3_2 + 1_1, 3_3 + 1_1, 2_1 + 2_1, 2_1 + 2_2, 2_2 + 2_1, 2_2 + 2_2; with a total of 2 x 10 = 20 parts.
(iii) 1_1 + 1_1 + 2_1, 1_1 + 1_1 + 2_2, 1_1 + 2_1 + 1_1, 1_1 + 2_2 + 1_1, 2_1 + 1_1 + 1_1, 2_2 + 1_1 + 1_1; with a total of 3 x 6 = 18 parts.
(iv) 1_1 + 1_1 + 1_1 + 1_1; with a total of 4 parts.
Hence, a(4) = 4 + 20 + 18 + 4 = 46.
(End)

R. P. Grimaldi, Compositions and the alternate Fibonacci numbers, Congressus Numerantium, 186, 2007, 81-96.

T. D. Noe, Table of n, a(n) for n = 1..200
A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
C. G. Bower, Transforms (2).
R. X. F. Chen and L. W. Shapiro, On sequences G(n) satisfying G(n)=(d+2)G(n-1)-G(n-2), J. Integer Seq. 10 (2007), Article #07.8.1; see Proposition 17.
Éva Czabarka, Rigoberto Flórez, and Leandro Junes, Some Enumerations on Non-Decreasing Dyck Paths, Electron. J. Combin., 22(1) (2015), #P1.3.
Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, J. Integer Seq., 18 (2015), Article #15.1.6.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math. 341 (10) (2018), 2789-2807.
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137 (1995), 155-176.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 6, 14-15, 17, 19.
Meghann Moriah Gibson, Combinatorics of compositions, Master of Science, Georgia Southern University, 2017.
Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color compositions, Discrete Mathematics 341 (2018), 3209-3226.
Milan Janjic, Hessenberg Matrices and Integer Sequences, J. Integer Seq. 13 (2010), Article #10.7.8.
N. J. A. Sloane, Transforms.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Partial sums of A038731. First differences of A001870.

Cf. A001629 (right-shifted inverse Binomial Transform), A023610 (inverse Binomial Transform of left-shifted sequence), A030279, A045623, A088305, A121462, A153294, A279282, A307415, A308723.

with(combinat): L[0]:=2: L[1]:=1: for n from 2 to 60 do L[n]:=L[n-1] +L[n-2] end do: seq(2*fibonacci(2*n)*1/5+(1/5)*n*L[2*n],n=1..30); # Emeric Deutsch, Jul 21 2008

Table[Sum[k Binomial[n+k-1,2k-1],{k,n}],{n,30}] (* or *) LinearRecurrence[ {6,-11,6,-1},{1,4,14,46},30] (* Harvey P. Dale, Aug 01 2011 *)

a(n)=(2*n*fibonacci(2*n+1)+(2-n)*fibonacci(2*n))/5

a(n) = -a(-n) = (2n * F(2n+1) + (2 - n) * F(2n))/5 with F(n) = A000045(n) (Fibonacci numbers).
G.f.: x * (1 - x)^2/(1 - 3*x + x^2)^2.
a(n) = Sum_{k = 1..n} k*C(n + k - 1, 2*k - 1).
a(n) = (2/5)*F(2*n) + (1/5)*n*L(2*n), where F(k) are the Fibonacci numbers (F(0)=0, F(1)=1) and L(k) are the Lucas numbers (L(0) = 2, L(1) = 1). - Emeric Deutsch, Jul 21 2008
a(0) = 1, a(1) = 4, a(2) = 14, a(3) = 46, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - Harvey P. Dale, Aug 01 2011
a(n) = ((3 - sqrt(5))^n*(5*n - 2*sqrt(5)) + (3 + sqrt(5))^n*(5*n + 2*sqrt(5)))/ (25*2^n). - Peter Luschny, Mar 07 2022
E.g.f.: exp(3*x/2)*(15*x*cosh(sqrt(5)*x/2) + sqrt(5)*(4 + 5*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

Name clarified using a comment of the author by Peter Luschny, Aug 03 2019

A279282 Self-composition of the cubes; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000578.

Original entry on oeis.org

0, 1, 16, 182, 1720, 14149, 106944, 760463, 5160488, 33756514, 214369376, 1328496947, 8065970016, 48125315989, 282851349184, 1640791635086, 9409099218712, 53408767286521, 300417148670400, 1676056809217283, 9282172245277448, 51062759750186170, 279196558362482192, 1518068927980989575

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for linear recurrences with constant coefficients, signature (20,-158,640,-1553,2920,-4806,5700,-6820,5700,-4806,2920,-1553,640,-158,20,-1).

Cf. A000578, A030267, A030279.

CoefficientList[Series[x (1 - x)^4 (1 + 4 x + x^2) (1 - 4 x + 29 x^2 - 84 x^3 + 152 x^4 - 84 x^5 + 29 x^6 - 4 x^7 + x^8)/((1 + x^2)^4 (1 - 5 x + x^2)^4), {x, 0, 23}], x]
LinearRecurrence[{20,-158,640,-1553,2920,-4806,5700,-6820,5700,-4806,2920,-1553,640,-158,20,-1},{0,1,16,182,1720,14149,106944,760463,5160488,33756514,214369376,1328496947,8065970016,48125315989,282851349184,1640791635086},30] (* Harvey P. Dale, Sep 27 2024 *)

G.f.: x*(1 - x)^4*(1 + 4*x + x^2)*(1 - 4*x + 29*x^2 - 84*x^3 + 152*x^4 - 84*x^5 + 29*x^6 - 4*x^7 + x^8)/((1 + x^2)^4*(1 - 5*x + x^2)^4).

A302356 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of squares (A000290).

Original entry on oeis.org

1, 8, 123, 3064, 107355, 4880896, 273564907, 18252720536, 1413701944431, 124714304306536, 12347969626724127, 1356049318451627812, 163596640499821625005, 21508738592360523314552, 3060986664449504902865167, 468816798653492762623354936, 76889830949170048691162162275

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			The initial coefficients of successive iterations of g.f. A(x) = x*(1 + x)/(1 - x)^3 are as follows:
n = 1: 0, (1),  4,    9,    16,      25,  ... g.f. A(x)
n = 2: 0,  1,  (8),  50,   276,    1397,  ... g.f. A(A(x))
n = 3: 0,  1,  12, (123), 1164,   10420,  ... g.f. A(A(A(x)))
n = 4: 0,  1,  16,  228, (3064),  39542,  ... g.f. A(A(A(A(x))))
n = 5: 0,  1,  20,  365,  6360, (107355), ... g.f. A(A(A(A(A(x)))))

N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000290, A030279, A119821, A302355.

Table[SeriesCoefficient[Nest[Function[x, x (1 + x)/(1 - x)^3], x, n], {x, 0, n}], {n, 17}]

A276644 Self-composition of the repunits; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A002275.

Original entry on oeis.org

0, 1, 22, 464, 9658, 199666, 4112922, 84558014, 1736623658, 35646098566, 731452470122, 15006822709814, 307859627711658, 6315326642698966, 129547066718721322, 2657377349777550614, 54509922224486463658, 1118139793621467673366, 22935894163202834676522, 470473020119757115115414

Offset: 0

Ilya Gutkovskiy, Sep 08 2016

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (33,-272,330,-100)

Cf. A000035, A002275, A063524.

Cf. A030267 (self-composition of the natural numbers), A030279 (self-composition of the squares), A030280 (self-composition of the triangular numbers).

I:=[0,1,22,464]; [n le 4 select I[n] else 33*Self(n-1)-272*Self(n-2)+330*Self(n-3)-100*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Sep 09 2016

LinearRecurrence[{33, -272, 330, -100}, {0, 1, 22, 464}, 20]

concat(0, Vec(x*(1-x)*(1-10*x)/((1-21*x+10*x^2)*(1-12*x+10*x^2)) + O(x^99))) \\ Altug Alkan, Sep 08 2016

O.g.f.: x*(1 - x)*(1 - 10*x)/((1 - 21*x + 10*x^2)*(1 - 12*x + 10*x^2)).
a(n) = 33*a(n-1) - 272*a(n-2) + 330*a(n-3) - 100*a(n-4) for n > 3.
a(n) = ((6 - sqrt(26))^n - (6 + sqrt(26))^n)/(18*sqrt(26)) + 10*(((21 + sqrt(401))/2)^n - ((21 - sqrt(401))/2)^n)/(9*sqrt(401)).
A000035(a(n)) = A063524(n).

A279284 Self-composition of the pentagonal numbers; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000326.

Original entry on oeis.org

0, 1, 10, 74, 469, 2662, 14115, 71360, 348143, 1652200, 7669883, 34969286, 157060011, 696514465, 3055404733, 13277356490, 57222978070, 244831062184, 1040760406476, 4398642943496, 18493603597214, 77388169532299, 322451025667910, 1338291853544522, 5534486308363461, 22812231761335189, 93741611639348947, 384122032722040412

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (12,-51,91,-75,66,-28,15,-3,1).

Cf. A000326, A030279, A030280.

CoefficientList[Series[x (1 - x)^3 (1 + 2 x) (1 - x + 7 x^2 - x^3)/(1 - 4 x + x^2 - x^3)^3, {x, 0, 25}], x]
LinearRecurrence[{12, -51, 91, -75, 66, -28, 15, -3, 1}, {0, 1, 10, 74, 469, 2662, 14115, 71360, 348143}, 26]

G.f.: x*(1 - x)^3*(1 + 2*x)*(1 - x + 7*x^2 - x^3)/(1 - 4*x + x^2 - x^3)^3.

a(n) = 12*a(n-1) - 51*a(n-2) + 91*a(n-3) - 75*a(n-4) + 66*a(n-5) - 28*a(n-6) + 15*a(n-7) - 3*a(n-8) + a(n-9).

cp's OEIS Frontend

A174654 Partial sums of A030279.

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A030267 Compose the natural numbers with themselves, A(x) = B(B(x)) where B(x) = x/(1-x)^2 is the generating function for natural numbers.

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A279282 Self-composition of the cubes; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000578.

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A302356 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of squares (A000290).

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A276644 Self-composition of the repunits; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A002275.

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Magma

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A279284 Self-composition of the pentagonal numbers; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000326.

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