A376729 -id:A376729 - cp's OEIS frontend

A108479 Antidiagonal sums of number triangle A086645.

1, 1, 2, 7, 17, 44, 117, 305, 798, 2091, 5473, 14328, 37513, 98209, 257114, 673135, 1762289, 4613732, 12078909, 31622993, 82790070, 216747219, 567451585, 1485607536, 3889371025, 10182505537, 26658145586, 69791931223, 182717648081

Offset: 0

Paul Barry, Jun 04 2005

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

Cf. A005252, A051286, A086645, A182890.
Cf. A381421, A382230, A382470, A382471, A382472, A382473, A382474.
Cf. A376729, A376730, A376731.

LinearRecurrence[{2,1,2,-1},{1,1,2,7},30] (* Harvey P. Dale, Jun 01 2021 *)

G.f.: (1 - x - x^2)/(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = 2*(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), 2*k).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*(n-2*k), j) * C(2*k, j).
a(n) = A005252(2*n). - Seiichi Manyama, Aug 11 2024

A376730 Expansion of (1 - x^3 - x^4)/((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 6, 1, 1, 15, 15, 2, 28, 70, 29, 46, 210, 211, 111, 496, 925, 586, 1067, 3005, 3123, 2821, 8100, 13024, 11068, 20385, 44068, 48604, 57325, 129261, 192224, 200585, 358806, 662117, 781433, 1055567, 2050819, 2941702, 3524140, 6067682, 10169037

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,2,2,0,-1,2,-1).

Cf. A108479, A376729, A376731.

Cf. A376724, A376727.

my(N=50, x='x+O('x^N)); Vec((1-x^3-x^4)/((1-x^3-x^4)^2-4*x^7))

a(n) = sum(k=0, n\3, binomial(2*k, 2*n-6*k));

a(n) = 2*a(n-3) + 2*a(n-4) - a(n-6) + 2*a(n-7) - a(n-8).

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,2*n-6*k).

A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 2, 2, 3, 10, 7, 28, 33, 64, 132, 170, 408, 578, 1119, 2002, 3194, 6310, 10021, 18666, 32353, 55450, 101443, 170672, 308744, 534820, 935936, 1663892, 2872669, 5111652, 8898082, 15641802, 27538647, 48049562, 84813451, 148219128, 260572901, 457451088

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Cf. A182890, A376724, A376725.
Cf. A376726, A376729.
Cf. A375278.

my(N=40, x='x+O('x^N)); Vec(1/((1-x^2-x^3)^2-4*x^5))

a(n) = sum(k=0, n\2, binomial(2*k+2, 2*n-4*k+1))/2;

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = (1/2) * Sum_{k=0..floor(n/2)} binomial(2*k+2,2*n-4*k+1).

A376726 Expansion of (1 + x^2 - x^3)/((1 + x^2 - x^3)^2 - 4*x^2).

Original entry on oeis.org

1, 0, 3, 1, 5, 10, 8, 35, 30, 85, 137, 201, 476, 616, 1357, 2172, 3735, 7193, 11213, 21782, 36064, 64095, 115130, 193769, 354737, 604049, 1074008, 1889968, 3273785, 5839608, 10106859, 17880785, 31325077, 54793282, 96710296, 168730043, 297336790, 520856765, 913684857

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Cf. A099511, A376727, A376728.

Cf. A376723, A376729.

my(N=40, x='x+O('x^N)); Vec((1+x^2-x^3)/((1+x^2-x^3)^2-4*x^2))

a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k+1));

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*k+1,2*n-4*k+1).

A376731 Expansion of (1 - x^4 - x^5)/((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 6, 1, 0, 1, 15, 15, 1, 1, 28, 70, 28, 2, 45, 210, 210, 46, 67, 495, 924, 496, 157, 1002, 3003, 3004, 1121, 1911, 8009, 12871, 8161, 4880, 18684, 43760, 43948, 23409, 41820, 126124, 184988, 133285, 113373, 324616, 647112, 657273, 454366

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,2,0,0,-1,2,-1).

Cf. A108479, A376729, A376730.

Cf. A376725, A376728.

my(N=60, x='x+O('x^N)); Vec((1-x^4-x^5)/((1-x^4-x^5)^2-4*x^9))

a(n) = sum(k=0, n\4, binomial(2*k, 2*n-8*k));

a(n) = 2*a(n-4) + 2*a(n-5) - a(n-8) + 2*a(n-9) - a(n-10).

a(n) = Sum_{k=0..floor(n/4)} binomial(2*k,2*n-8*k).

A382300 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(2k,2n-4*k).

Original entry on oeis.org

1, 0, 2, 2, 3, 18, 7, 60, 65, 144, 356, 410, 1272, 1722, 3743, 7202, 11482, 25566, 40421, 81610, 147169, 259810, 507267, 867792, 1659112, 2961860, 5362592, 9940420, 17583485, 32564548, 58228386, 105606458, 191831767, 343313042, 625086891, 1119760040, 2023087045

Offset: 0

Seiichi Manyama, Mar 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,4,4,-6,-4,-2,-4,-5,8,-6,4,-1).

Cf. A376729, A382494, A382495.
Cf. A381421, A382496.
Cf. A034839, A375218.

R:=PowerSeriesRing(Rationals(), 37); Coefficients(R!( ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2)); // Vincenzo Librandi, May 11 2025

Table[Sum[(k+1)*Binomial[2*k,2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n\2, (k+1)*binomial(2*k, 2*n-4*k));

my(N=1, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2.

a(n) = 4*a(n-2) + 4*a(n-3) - 6*a(n-4) - 4*a(n-5) - 2*a(n-6) - 4*a(n-7) - 5*a(n-8) + 8*a(n-9) - 6*a(n-10) + 4*a(n-11) - a(n-12).

A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2k,2n-4*k).

Original entry on oeis.org

1, 0, 3, 3, 6, 36, 16, 150, 165, 430, 1071, 1365, 4453, 6258, 14841, 29169, 49941, 115356, 190091, 404811, 750792, 1393956, 2808438, 4988268, 9905746, 18207126, 34231566, 65278964, 119255889, 227648406, 418394087, 782045001, 1457704212, 2681909302

Offset: 0

Seiichi Manyama, Mar 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,6,6,-15,-18,5,12,-3,32,12,-6,-4,18,-33,26,-15,6,-1).

Cf. A376729, A382300, A382495.

Cf. A034839, A377146, A382230.

[&+[Binomial(k+2, 2)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..41]]; // Vincenzo Librandi, May 11 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)

a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(2*k, 2*n-4*k));

my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x^2-x^3)^(3-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^3.

a(n) = 6*a(n-2) + 6*a(n-3) - 15*a(n-4) - 18*a(n-5) + 5*a(n-6) + 12*a(n-7) - 3*a(n-8) + 32*a(n-9) + 12*a(n-10) - 6*a(n-11) - 4*a(n-12) + 18*a(n-13) - 33*a(n-14) + 26*a(n-15) - 15*a(n-16) + 6*a(n-17) - a(n-18).

A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2k,2n-4*k).

Original entry on oeis.org

1, 0, 4, 4, 10, 60, 30, 300, 335, 1000, 2506, 3500, 11879, 17304, 44220, 88592, 161865, 385704, 660964, 1475100, 2807956, 5459860, 11313094, 20816004, 42774780, 80798128, 157292750, 307887904, 579776799, 1138007940, 2146348214, 4126143900, 7878910238, 14878269368

Offset: 0

Seiichi Manyama, Mar 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0,8,8,-28,-40,28,72,2,16,20,-80,-114,56,-68,0,35,40,-96,128,-110,64,-28,8,-1).

Cf. A376729, A382300, A382494.

Cf. A034839, A382470.

[&+[Binomial(k+3, 3)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, May 12 2025

Table[Sum[Binomial[k+3,3]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,33}] (* Vincenzo Librandi, May 12 2025 *)

a(n) = sum(k=0, n\2, binomial(k+3, 3)*binomial(2*k, 2*n-4*k));

my(N=3, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x^2-x^3)^(4-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^4.

a(n) = 8*a(n-2) + 8*a(n-3) - 28*a(n-4) - 40*a(n-5) + 28*a(n-6) + 72*a(n-7) + 2*a(n-8) + 16*a(n-9) + 20*a(n-10) - 80*a(n-11) - 114*a(n-12) + 56*a(n-13) - 68*a(n-14) + 35*a(n-16) + 40*a(n-17) - 96*a(n-18) + 128*a(n-19) - 110*a(n-20) + 64*a(n-21) - 28*a(n-22) + 8*a(n-23) - a(n-24).

A376787 Expansion of (1 - x^2 + x^3)/((1 - x^2 + x^3)^2 - 4*x^3).

Original entry on oeis.org

1, 0, 1, 3, 1, 10, 6, 21, 36, 43, 127, 139, 340, 540, 881, 1832, 2653, 5427, 8829, 15550, 28642, 46805, 87756, 147575, 262751, 465591, 797864, 1437816, 2471553, 4383696, 7689305, 13402819, 23752217, 41305842, 72916606, 127708213, 223809012, 394045411

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Cf. A376716, A376788.

Cf. A376723, A376726, A376729.

CoefficientList[Series[(1-x^2+x^3)/((1-x^2+x^3)^2-4x^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,2,2,-1,2,-1},{1,0,1,3,1,10},40] (* Harvey P. Dale, Aug 11 2025 *)

my(N=40, x='x+O('x^N)); Vec((1-x^2+x^3)/((1-x^2+x^3)^2-4*x^3))

a(n) = sum(k=0, n\2, binomial(2*k+1, 2*n-4*k));

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*k+1,2*n-4*k).

cp's OEIS Frontend

A108479 Antidiagonal sums of number triangle A086645.

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A376730 Expansion of (1 - x^3 - x^4)/((1 - x^3 - x^4)^2 - 4*x^7).

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A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

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A376726 Expansion of (1 + x^2 - x^3)/((1 + x^2 - x^3)^2 - 4*x^2).

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A376731 Expansion of (1 - x^4 - x^5)/((1 - x^4 - x^5)^2 - 4*x^9).

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A382300 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(2*k,2*n-4*k).

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A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).

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A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2*k,2*n-4*k).

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A382300 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(2k,2n-4*k).

A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2k,2n-4*k).

A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2k,2n-4*k).