A040977 -id:A040977 - cp's OEIS frontend

A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

1, 5, 6, 20, 27, 77, 112, 294, 450, 1122, 1782, 4290, 7007, 16445, 27456, 63206, 107406, 243542, 419900, 940576, 1641486, 3640210, 6418656, 14115100, 25110020, 54826020, 98285670, 213286590, 384942375

Offset: 1

Luciano Ancora, Jul 05 2015

The term "stepped path" in the name field is the same used in A001405.

Interleaving of terms of the sequences A220101 and A129869. - Michel Marcus, Jul 05 2015

			The array of k-th partial sums of squares begins:
[1], [5],  14,   30,    55,     91,  ...  A000330
1,   [6], [20],  50,   105,    196,  ...  A002415
1,    7,  [27], [77],  182,    378,  ...  A005585
1,    8,   35, [112], [294],   672,  ...  A040977
1,    9,   44,  156,  [450], [1122], ...  A050486
1,   10,   54,  210,   660,  [1782], ...  A053347
This is essentially A110813 without its first two columns.

Cf. A000330, A002415, A005585, A040977, A050486, A053347.

Table[DifferenceRoot[Function[{a, n}, {(-9168 - 14432*n - 8412*n^2 - 2152*n^3 - 204*n^4)*a[n] +(-1332 - 1902*n - 792*n^2 - 102*n^3)*a[1 + n] + (2100 + 3884*n + 2493*n^2 + 640*n^3 + 51*n^4)*a[2 + n] == 0, a[1] == 1 , a[2] == 5}]][n], {n, 29}]

Conjecture: -(n+5)*(13*n-11)*a(n) +(8*n^2+39*n-35)*a(n-1) +2*(26*n^2+48*n+25)*a(n-2) -4*(8*n+5)*(n-1)*a(n-3)=0. - R. J. Mathar, Jul 16 2015

A266561 12-dimensional square numbers.

Original entry on oeis.org

1, 14, 104, 546, 2275, 8008, 24752, 68952, 176358, 419900, 940576, 1998724, 4056234, 7904456, 14858000, 27041560, 47805615, 82317690, 138389160, 227613750, 366913365, 580610160, 903171360, 1382805840, 2086129500, 3104160696, 4559958144, 6618272584

Offset: 0

Antal Pinter, Dec 31 2015

2*a(n) is number of ways to place 11 queens on an (n+11) X (n+11) chessboard so that they diagonally attack each other exactly 55 times. The maximal possible attack number, p=binomial(k,2)=55 for k=11 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph.

Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
OEIS Wiki, Square hyperpyramidal numbers, line d=12 of first table.

Cf. A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

[Binomial(n+11,11)*(n+6)/6: n in [0..40]]; // Vincenzo Librandi, Jan 01 2016

CoefficientList[Series[(1 + x)/(1 - x)^13, {x, 0, 33}], x] (* Vincenzo Librandi, Jan 01 2016 *)

a(n) = binomial(n+11,11)*(n+6)/6.
a(n) = 2*binomial(n+12,12) - binomial(n+11,11).
a(n) = binomial(n+11,11) + 2*binomial(n+11,12) for n>0.
G.f.: (1+x)/(1-x)^13. - Vincenzo Librandi, Jan 01 2016

A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

Original entry on oeis.org

1, 1, -2, 3, -4, 2, 15, -18, 9, -2, 105, -120, 60, -16, 2, 945, -1050, 525, -150, 25, -2, 10395, -11340, 5670, -1680, 315, -36, 2, 135135, -145530, 72765, -22050, 4410, -588, 49, -2, 2027025, -2162160, 1081080, -332640, 69300, -10080, 1008, -64, 2

Offset: 0

Johannes W. Meijer, May 31 2018

The function f(u, p) = (1/2)*(1+1/(sqrt(1-u^2))) * exp(p*sqrt(1-u^2)) was found while studying the Fresnel-Kirchhoff and the Rayleigh-Sommerfeld theories of diffraction, see the Meijer link.
The Taylor expansion of f(u, p) leads to the number triangle T(n, k), see the example section.
Normalization of the triangle terms, dividing the T(n, k) by T(n-k, 0), leads to A084534.
The row sums equal A003436, n >= 2, respectively A231622, n >= 1.

			The first few terms of the Taylor expansion of f(u; p) are:
f(u, p) = exp(p) * (1 + (1-2*p) * u^2/4 + (3-4*p+2*p^2) * u^4/16 + (15-18*p+9*p^2-2*p^3) * u^6/96 + (105-120*p+60*p^2-16*p^3+2*p^4) * u^8/768 + ... )
The first few rows of the T(n, k) triangle are:
n=0:     1
n=1:     1,     -2
n=2:     3,     -4,    2
n=3:    15,    -18,    9,    -2
n=4:   105,   -120,   60,   -16,   2
n=5:   945,  -1050,  525,  -150,  25,  -2
n=6: 10395, -11340, 5670, -1680, 315, -36, 2

J. W. Goodman, Introduction to Fourier Optics, 1996.
A. Papoulis, Systems and Transforms with Applications in Optics, 1968.

Andrew Howroyd, Rows n=0..50 of triangle, flattened
M. J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Optics Communications, Vol. 25, nr. 1, pp. 26-30, 1978.
H. J. Butterweck, General theory of linear, coherent optical data processing systems, Journal of the Optical Society of America, Vol. 67, nr. 1, pp. 60-70, 1977.
J. W. Meijer, A note on optical diffraction, 1979.

Cf. A003436, A231622, A032184, A084534, A127674, A132062, A001497.
Cf. Related to the left hand columns: A001147, A001193, A261065.
Cf. Related to the right hand columns: A280560, A162395, A006011, A040977, A053347, A054334, A266561.

[[n le 0 select 1 else (-1)^k*2^(k-n+1)*Factorial(2*n-k-1)*Binomial(n, k)/Factorial(n-1): k in [0..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018

T := proc(n, k): if n=0 then 1 else (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!) fi: end: seq(seq(T(n, k), k=0..n), n=0..8);

Table[If[n==0 && k==0,1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!)], {n, 0, 10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 08 2018 *)

T(n,k) = {if(n==0, 1, (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 08 2018

T(n, k) = (-1)^k*2^(k-n+1)*n*(2*n-k-1)!/(k!*(n-k)!), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*A001147(n-k)*A084534(n, k), n >= 0 and 0 <= k <= n.
T(n, k) = 2^(2*(k-n)+1)*A001147(n-k)*A127674(n, n-k), n > 0 and 0 <= k <= n, T(0, 0) = 1.
T(n, k) = (-1)^k*(A001497(n, k) + A132062(n, k)), n >= 1, T(0,0) = 1.

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.

Original entry on oeis.org

1, 3, 4, 6, 16, 10, 10, 40, 50, 20, 15, 80, 150, 120, 35, 21, 140, 350, 420, 245, 56, 28, 224, 700, 1120, 980, 448, 84, 36, 336, 1260, 2520, 2940, 2016, 756, 120, 45, 480, 2100, 5040, 7350, 6720, 3780, 1200, 165, 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220

Offset: 0

Luc Rousseau, Aug 14 2021

This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.

Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

			T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
The triangle T begins:
n \ k  0   1    2     3     4     5     6     7     8    9  10 ...
0:     1
1:     3   4
2:     6  16   10
3:    10  40   50    20
4:    15  80  150   120    35
5:    21 140  350   420   245    56
6:    28 224  700  1120   980   448    84
7:    36 336 1260  2520  2940  2016   756   120
8:    45 480 2100  5040  7350  6720  3780  1200   165
9:    55 660 3300  9240 16170 18480 13860  6600  1815  220
10:   66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
... - _Wolfdieter Lang_, Sep 30 2021

Luc Rousseau, Illustration: the A347056 triangle in a sequence of contiguous triangles.

Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
From Wolfdieter Lang, Sep 30 2021: (Start)
Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)

T(p,n,k)=if(n==0&&p==0,1,((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
for(n=0,9,for(k=0,n,print1(T(3,n,k),", ")))

T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.

G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - Wolfdieter Lang, Sep 30 2021

A339356 Maximum number of copies of a 123456 permutation pattern in an alternating (or zig-zag) permutation of length n + 9.

Original entry on oeis.org

16, 32, 144, 256, 688, 1120, 2352, 3584, 6496, 9408, 15456, 21504, 32928, 44352, 64416, 84480, 117744, 151008, 203632, 256256, 336336, 416416, 534352, 652288, 821184, 990080, 1226176, 1462272, 1785408, 2108544, 2542656, 2976768, 3550416, 4124064, 4870992, 5617920, 6577648

Offset: 1

Lara Pudwell, Dec 01 2020

The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous.

			a(1) = 16. The alternating permutation of length 1+9=10 with the maximum number of copies of 123456 is 132547698(10). The sixteen copies are 12468(10), 12469(10), 12478(10), 12479(10), 12568(10), 12569(10), 12578(10), 12579(10), 13468(10), 13469(10), 13478(10), 13479(10), 13568(10), 13569(10), 13578(10), and 13579(10).

Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society. 67.7 (2020), 994-1001.
Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Cf. A168380.

a(2n) = 32*A040977(n-1) = 64*C(n+5,6) - 32*C(n+4,5).
a(2n-1) = 16*A259181(n) = (2*n*(n + 1)*(n + 2)*(n + 3)*(2*n^2 + 6*n + 7))/45.
From Chai Wah Wu, Jul 06 2025: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n > 12.
G.f.: x*(-16*x^2 - 16)/((x - 1)^7*(x + 1)^5). (End)

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 10, 16, 6, 1, 20, 70, 76, 24, 1, 35, 224, 496, 428, 120, 1, 56, 588, 2260, 3808, 2808, 720, 1, 84, 1344, 8140, 23008, 32152, 21096, 5040, 1, 120, 2772, 24772, 107328, 245560, 298688, 178848, 40320

Offset: 0

Michael Somos, Mar 15 2023

This sequence is essentially A204024 except for extra row, alternating signs and reversed rows.

The sequence of polynomials a(m) satisfies a(m)*a(m-2) = a(m-1) * (a(m-1) + x*a(m-2) + a(m-3)) - a(m-2)^2 for all m > 3.

			a(3) = 1, a(4) = 1 + x, a(5) = 1 + 4*x + 2*x^2.
Triangular array T(n, k) starts:
n\k | 0   1   2   3   4   5
--- + - --- --- --- --- ---
 0  | 1
 1  | 1   1
 2  | 1   4   2
 3  | 1  10  16   6
 4  | 1  20  70  76  24
 5  | 1  35 224 496 428 120

Cf. A000292, A040977, A058797, A093986, A204024.

T[ n_, k_] := If[ n<0, 0, Module[{a = Table[1, n+3], x}, Do[ a[[m]] = a[[m-1]] *(a[[m-1]] + x*a[[m-2]] + a[[m-3]])/a[[m-2]] - a[[m-2]] //Factor//Expand, {m, 4, n+3}]; Coefficient[ a[[n+3]], x, k]]];

{T(n, k) = if( n<0, 0, my(a = vector(n+3, i, 1)); for(m = 4, n+3, a[m] = a[m-1]*(a[m-1] + 'x*a[m-2] + a[m-3])/a[m-2] - a[m-2]); polcoeff( a[n+3], k))};

If x=1, then a(n) = A058797(n+2) = Sum_{k=0..n} T(n, k).
If x=2, then a(n) = A093986(n+2).
T(n, n) = n!, T(n, 0) = 1, T(n, 1) = A000292(n). T(n, 2) = 2*A040977(n-2).

cp's OEIS Frontend

A259775 Stepped path in P(k,n) array of k-th partial sums of squares (A000290).

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A266561 12-dimensional square numbers.

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)*(1+1/(sqrt(1-u^2)))*exp(p*sqrt(1-u^2)).

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A347056 Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.

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A339356 Maximum number of copies of a 123456 permutation pattern in an alternating (or zig-zag) permutation of length n + 9.

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A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.

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A305402 A number triangle T(n,k) read by rows for 0<=k<=n, related to the Taylor expansion of f(u, p) = (1/2)(1+1/(sqrt(1-u^2)))exp(p*sqrt(1-u^2)).

A347056 Triangle read by rows: T(n,k) = (n+1)(n+2)(k+3)*binomial(n,k)/6, 0 <= k <= n.

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.