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x/(x+1)","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:             441,              504,              144;\n  1:            9261,            15876,             9072,              1728;\n  2:          194481,           444528,           381024,            145152,             20736;\n  3:         4084101,         11668860,         13335840,           7620480,           2177280,           248832;\n  4:        85766121,        294055272,        420078960,         320060160,         137168640,         31352832,          2985984;\n  5:      1801088541,       7204354164,      12350321424,       11762210880,        6721263360,       2304433152,        438939648,         35831808;\n  6:     37822859361,     172904499936,     345808999872,      395210285568,      282293061120,     129048256512,      36870930432,       6019743744,       429981696;\n  7:    794280046581,    4084868810988,    9336842996544,    12449123995392,    10670677710336,    6097530120192,    2322868617216,     568865783808,     81266540544,      5159780352;\n  8:  16679880978201,   95313605589720,  245092128659280,   373473719861760,   373473719861760,  256096265048064,  121950602403840,   39820604866560,   8532986757120,   1083553873920,    61917364224;\n  9: 350277500542221, 2201744289122532, 6290697968921520, 10784053661008320, 12324632755438080, 9859706204350464, 5634117831057408, 2299639931043840, 657039980298240, 125150472437760, 14302911135744, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 504*x + 144*x^2.","mathematica":"P[0, x_] := 441 + 504*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[(21 + 12*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] = 441 + 504*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 12*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-615df6ea54d3bbef","steps":["basis A013609","basis A013609"],"edges":["e-f3e6c91e6b1bdc2d","e-09fc2301440d61ef"],"text":"A038329 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038329 \u2192 basis A013609 \u2192 basis A013609","chain":null}