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n.","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["25","15"],["375","585","135"]],"card_rows":[["1"],["25","15"],["375","585","135"],["4375","11655","6615","945"]],"row_sums_preview":["1","40","1095","23590","436030","7254072","111744570","1622994780"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","basis A008288"],"edges":["e-0c010b1c5e20b0cb","e-1abdf6dc0212b840"],"text":"A178821 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A178821 \u2192 basis A013610 \u2192 basis A008288","chain":null}},{"id":"l-d2c52e74c08fbd52","kind":"fingerprint_related","node_a_id":"n-7779a9e47344358a","node_b_id":"n-056fa906003b296d","fingerprint":"f6-0fee5d4c7376c95e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0fee5d4c7376c95e"},"other_id":"n-7779a9e47344358a","other":{"id":"n-7779a9e47344358a","name":"basis_map_A008288 of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-0fee5d4c7376c95e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["10","6"],["50","78","18"]],"card_rows":[["2"],["10","6"],["50","78","18"],["250","666","378","54"]],"row_sums_preview":["2","16","146","1348","12458","115144","1064234","9836332"],"regions":["[-1564556493/268435456, -46056243/268435456]"],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","basis A008288"],"edges":["e-28b777b3c0b25332","e-9556e50b57d29665"],"text":"A028326 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A028326 \u2192 basis A013610 \u2192 basis A008288","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:                 3600,                  5616,                   1296;\n  1:               216000,                575424,                 326592,                  46656;\n  2:             12960000,              49579776,               48708864,               16236288,                1679616;\n  3:            777600000,            3895962624,             5654707200,             3191270400,              745749504,               60466176;\n  4:          46656000000,          289458302976,           566527758336,           469298147328,           182124122112,            32651735040,             2176782336;\n  5:        2799360000000,        20719850471424,         51549108092928,         57663447810048,         32497909088256,          9534306631680,          1384433565696,            78364164096;\n  6:      167961600000000,      1444579835314176,       4385285711659008,       6268448166838272,       4746186548379648,       2002465606533120,        470498441232384,         57362568118272,         2821109907456;\n  7:    10077696000000000,     98764062607540224,     355089133600505856,     623653781483814912,     602457424436920320,     339379521866956800,     113735867028996096,      22241630510383104,      2335879003373568,      101559956668416;\n  8:   604661760000000000,   6651342763380965376,   27692240540380692480,   58038004521448243200,   68931420764526084096,   49363149232045817856,   21959023004293791744,    6083441404438118400,   1017630765817528320,    93841399961616384,    3656158440062976;\n  9: 36279705600000000000, 442613930001143169024, 2096934267535421865984, 5127450099634082414592, 7281487249404170600448, 6399590778883119513600, 3604463609639098908672, 1318639852748953092096, 310729593504072204288, 45380238558061658112, 3729281608864235520, 131621703842267136;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1440-432*x)*P(n-2,x) + (84+36*x)*P(n-1,x) for n >= 2, with P(0,x) = 3600 + 5616*x + 1296*x^2, P(1,x) = 216000 + 575424*x + 326592*x^2 + 46656*x^3.","mathematica":"P[0, x_] := 3600 + 5616*x + 1296*x^2;\nP[1, x_] := 216000 + 575424*x + 326592*x^2 + 46656*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1440 - 432*x)*P[n - 2, x] + (84 + 36*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 5616*x + 1296*x^2; v[2] = 216000 + 575424*x + 326592*x^2 + 46656*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(1440 - 432*x)*v[i - 2] + (84 + 36*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","basis A008288"],"edges":["e-85e8843cd1dca083","e-d120a6d023071b51"],"text":"A038338 \u2192 basis A013610 \u2192 basis A008288"},"headline":"A038338 \u2192 basis A013610 \u2192 basis A008288","chain":null}