A002418 -id:A002418 - cp's OEIS frontend

A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

1, 5, 2, 15, 9, 3, 35, 25, 13, 4, 70, 55, 35, 17, 5, 126, 105, 75, 45, 21, 6, 210, 182, 140, 95, 55, 25, 7, 330, 294, 238, 175, 115, 65, 29, 8, 495, 450, 378, 294, 210, 135, 75, 33, 9, 715, 660, 570, 462, 350, 245, 155, 85, 37, 10, 1001, 935, 825, 690, 546

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal:  A002418
Antidiagonal sums:  A005585
row 1,  (1,3,6,...)**(1,2,3,...):  A000332
row 2,  (1,3,6,...)**(2,3,4,...):  A005582
row 3,  (1,3,6,...)**(3,4,5,...):  A095661
row 4,  (1,3,6,...)**(4,5,6,...):  A095667
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....5....15...35....70....126
2....9....25...55....105...182
3....13...35...75....140...238
4....17...45...95....175...294
5....21...55...115...210...350

Cf. A213500, A213548.

b[n_] := n (n + 1)/2; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213550 *)
d = Table[t[n, n], {n, 1, 40}] (* A002418 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A005585 *)

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n-(n-1)*x and g(x) = (1 - x)^5.

A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

Original entry on oeis.org

1, 7, 1, 18, 8, 1, 34, 26, 9, 1, 55, 60, 35, 10, 1, 81, 115, 95, 45, 11, 1, 112, 196, 210, 140, 56, 12, 1, 148, 308, 406, 350, 196, 68, 13, 1, 189, 456, 714, 756, 546, 264, 81, 14, 1, 235, 645, 1170, 1470, 1302, 810, 345, 95, 15, 1, 286, 880, 1815, 2640, 2772, 2112, 1155, 440, 110, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

The leftmost column contains the heptagonal numbers A000566.
The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.
Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.

			First few rows of the triangle are:
  1;
  7, 1;
  18, 8, 1;
  34, 26, 9, 1;
  55, 60, 35, 10, 1;
  81, 115, 95, 45, 11, 1;
  112, 196, 210, 140, 56, 12, 1;
Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1966, p. 189.

Cf. A000566, A002413, A002418, A027800, A051946, A050484.

Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.

A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n,k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1,k-1)+procname(n-1,k) ; fi; end: seq(seq(A125234(n,k),k=0..n-1),n=1..16) ; # R. J. Mathar, Sep 09 2009

A000566[n_] := n(5n-3)/2;
T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.

Edited and extended by R. J. Mathar, Sep 09 2009

A261721 Fourth-dimensional figurate numbers.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 20, 35, 1, 8, 25, 50, 70, 1, 9, 30, 65, 105, 126, 1, 10, 35, 80, 140, 196, 210, 1, 11, 40, 95, 175, 266, 336, 330, 1, 12, 45, 110, 210, 336, 462, 540, 495, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 1, 14, 55, 140, 280, 476, 714, 960, 1155, 1210, 1001, 1

Offset: 1

Gary W. Adamson, Aug 30 2015

Generating polygons for the sequences are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ... .
n-th row sequence is the binomial transform of the fourth row of Pascal's triangle (1,n) followed by zeros; and the fourth partial sum of (1, n, n, n, ...).
n-th row sequence is the binomial transform of:
((n-1) * (0, 1, 3, 3, 1, 0, 0, 0) + (1, 4, 6, 4, 1, 0, 0, 0)).
Given the n-th row of the array (1, b, c, d, ...), the next row of the array is (1, b, c, d, ...) + (0, 1, 5, 15, 35, ...)

			The array as shown in A257200:
  1,  5, 15,  35,  70, 126, 210,  330, ... A000332
  1,  6, 20,  50, 105, 196, 336,  540, ... A002415
  1,  7, 25,  65, 140, 266, 462,  750, ... A001296
  1,  8, 30,  80, 175, 336, 588,  960, ... A002417
  1,  9, 35,  95, 210, 406, 714, 1170, ... A002418
  1, 10, 40, 110, 245, 476, 840, 1380, ... A002419
  ...
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of the fourth row of Pascal's triangle (1,3) followed by zeros: (1, 6, 12, 10, 3, 0, 0, 0, ...); and the fourth partial sum of (1, 3, 3, 3, 0, 0, 0).
(1, 7, 25, 65, 140, ...) is the third row of the array and is the binomial transform of: (2 * (0, 1, 3, 3, 1, 0, 0, 0, ...) + (1, 4, 6, 4, 1, 0, 0, 0, ...)); that is, the binomial transform of (1, 6, 12, 10, 3, 0, 0, 0, ...).
Row 2 of the array is (1, 5, 15, 35, 70, ...) + (0, 1, 5, 15, 35, ...), = (1, 6, 20, 50, 105, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 195 (Table 80).

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index to sequences related to pyramidal numbers

Cf. A257200, A261720 (pyramidal numbers), A000332, A002415, A001296, A002417, A002418, A002419.
Similar to A080852 but without row n=0.
Main diagonal gives A256859.

A:= (n, k)-> binomial(k+3, 3) + n*binomial(k+3, 4):
seq(seq(A(d-k, k), k=0..d-1), d=1..13);  # Alois P. Heinz, Aug 31 2015

row[1] = LinearRecurrence[{5, -10, 10, -5, 1}, {1, 5, 15, 35, 70}, m = 10];
row1 = Join[{0}, row[1] // Most]; row[n_] := row[n] = row[n-1] + row1;
Table[row[n-k+1][[k]], {n, 1, m}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 26 2016 *)

A(n, k) = binomial(k+3, 3) + n*binomial(k+3, 4)
table(n, k) = for(x=1, n, for(y=0, k-1, print1(A(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns as follows: */
table(6, 8) \\ Felix Fröhlich, Jul 28 2016

G.f. for row n: (1 + (n-1)*x)/(1 - x)^5.
A(n,k) = C(k+3,3) + n * C(k+3,4) = A080852(n,k).
E.g.f. as array: exp(y)*(exp(x)*(24 + 24*(3 + x)*y + 36*(1 + x)*y^2 + 4*(1 + 3*x)*y^3 + x*y^4) - 4*(6 + 18*y + 9*y^2 + y^3))/24. - Stefano Spezia, Aug 15 2024

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 10, 45, 185, 710, 2766, 10270, 37444, 132765, 462327, 1579563, 5311361, 17584084, 57414594, 185032557, 589183035, 1854974066, 5778722817, 17823440534, 54458411332, 164917654587, 495219323844, 1475145786950, 4360576440676, 12796007418881, 37287660835368, 107930276062786

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Euler transform of A002418.

N. J. A. Sloane, Transforms

Cf. A000391, A002418, A278769, A279218, A305653, A317017, A317019, A317021.

a:=series(mul(1/(1-x^k)^((5*k-1)*binomial(k+2,3)/4),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^((5 k - 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 4 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (5 d - 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002418(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 4*x^k)/(k*(1 - x^k)^5)).
a(n) ~ (5/7)^(703/8640)/(2 * 3^(2143/2880) * n^(5023/8640) * Pi^(17/1440)) * exp(-1/144 + (1/12-Zeta'(-1))/12 - (21 * Zeta(3))/(400 * Pi^2) + (62921 * Zeta(5))/(80000 * Pi^4) - (194481 * Zeta(3) * Zeta(5)^2)/(50 * Pi^12) - (200120949 * Zeta(5)^3)/(1250 * Pi^14) + (28594081676916 * Zeta(5)^5)/(3125 * Pi^24) + (7 * Zeta'(-3))/12 + ((-343 * (7/5)^(1/6) * Pi)/(96000 * sqrt(3)) + (147 * (7/5)^(1/6) * sqrt(3) * Zeta(3) * Zeta(5))/(10 * Pi^7) + (1058841 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^2)/(2000 * Pi^9) - (18211006359 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^4)/(500 * Pi^19)) * n^(1/6) + (-((7/5)^(1/3) * Zeta(3))/(4 * Pi^2) - (1029 * (7/5)^(1/3) * Zeta(5))/(200 * Pi^4) + (10890936 * (7/5)^(1/3) * Zeta(5)^3)/(25 * Pi^14)) * n^(1/3) + ((7 * sqrt(7/15) * Pi)/120 - (9261 * sqrt(21/5) * Zeta(5)^2)/(5 * Pi^9)) * sqrt(n) + ((63 * (7/5)^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * sqrt(3) * Pi)/(5^(5/6) * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A322652 Numbers that are sums of consecutive heptagonal pyramidal numbers (A002413).

Original entry on oeis.org

0, 1, 8, 9, 26, 34, 35, 60, 86, 94, 95, 115, 175, 196, 201, 209, 210, 308, 311, 371, 397, 405, 406, 456, 504, 619, 645, 679, 705, 713, 714, 764, 880, 960, 1075, 1101, 1135, 1161, 1166, 1169, 1170, 1409, 1508, 1525, 1605, 1720, 1780, 1806, 1814, 1815, 1911, 1981, 2046, 2289, 2380

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number

Cf. A002413, A002418, A321450, A322468, A322479, A322636, A322640, A322651, A322653.

imax = 55;
A002413 = LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, imax];
Join[{0}, Table[A002413[[i ;; j]] // Total, {i, 1, imax}, {j, i, imax}] // Flatten // Union][[;; imax]] (* Jean-François Alcover, Nov 11 2024 *)

A366036 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 4 * A(x)).

Original entry on oeis.org

0, 1, 9, 127, 2165, 40914, 824859, 17383720, 378373437, 8440227235, 191938302578, 4433259845898, 103716352560119, 2452629475989840, 58529969579982600, 1407775987050271920, 34092047564798908045, 830565580516900384329, 20342106952028722530603, 500573735323751221019425, 12370242700776737398052970

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for 4-dimensional figurate numbers A002418 (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002294, A002418, A064088, A365668, A365755, A365817, A366016, A366035, A366037.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 4^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 4^k for n > 0.

a(n) ~ sqrt(34*sqrt(6) - 81) * 2^(n - 11/4) * 3^(n - 5/4) * (3/2 - 1/sqrt(6))^(5*n) / (sqrt(Pi) * n^(3/2) * (3*sqrt(6) - 7)^n). - Vaclav Kotesovec, Sep 27 2023

A234277 a(n) = binomial(floor(n/2),4) + (ceiling(n/2)-3)*binomial(floor(n/2),3).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 5, 9, 25, 35, 75, 95, 175, 210, 350, 406, 630, 714, 1050, 1170, 1650, 1815, 2475, 2695, 3575, 3861, 5005, 5369, 6825, 7280, 9100, 9660, 11900, 12580, 15300, 16116, 19380, 20349, 24225, 25365, 29925, 31255, 36575, 38115, 44275, 46046, 53130, 55154, 63250, 65550, 74750, 77350, 87750, 90675

Offset: 0

N. J. A. Sloane, Dec 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica 33 (2013), no. 5, 591--612. MR3132928.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A000292, A002418.

CoefficientList[Series[-x^7 (4 x + 1)/((x - 1)^5 (x + 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi Feb 01 2014 *)
LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{0,0,0,0,0,0,0,1,5},60] (* Harvey P. Dale, Jun 27 2020 *)

Vec(-x^7*(4*x+1)/((x-1)^5*(x+1)^4) + O(x^100)) \\ Colin Barker, Jan 02 2014

G.f.: -x^7*(4*x+1) / ((x-1)^5*(x+1)^4). - Colin Barker, Jan 02 2014

a(n) = (2*n - 1 + (-1)^n)*(2*n - 5 + (-1)^n)*(2*n - 9 + (-1)^n)*(10*n - 57 - 3*(-1)^n)/6144. - Luce ETIENNE, Nov 18 2017

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

G.f.: x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A085787.
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017

cp's OEIS Frontend

A213550 Rectangular array: (row n) = b**c, where b(h) = h*(h+1)/2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A261721 Fourth-dimensional figurate numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A322652 Numbers that are sums of consecutive heptagonal pyramidal numbers (A002413).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A366036 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 4 * A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A234277 a(n) = binomial(floor(n/2),4) + (ceiling(n/2)-3)*binomial(floor(n/2),3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

Views

Author

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

A287143 Expansion of x(1 + 3x + x^2)/((1 - x)^5*(1 + x)^4).