{"id":"n-ab60eb3242173f47","revision":0,"first_index":0,"fingerprint":"f6-f34dcee26c4e5272","name":"reversal of A convolution triangle of numbers obtained from A036224.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","42","336"],["1","63","1113","4536"],["1","84","2331","23184","54432"],["1","105","3990","65205","412272","598752"],["1","126","6090","139860","1518048","6531840","6158592"],["1","147","8631","256410","4010769","30912840","94618368","60046272"],["1","168","11613","424116","8723925","99302112","567362016","1274921856","560431872"],["1","189","15036","652239","16685487","253790901","2193126624","9573741216","16176101760","5043886848"],["1","210","18900","950040","29117907","557085438","6534131760","44153890560","150628802688","195030291456","44019376128"],["1","231","23205","1326780","47438118","1096591041","16358573673","152347155480","822902685696","2232875211840","2250184914432","374164697088"],["1","252","27951","1791720","73257534","1988496216","36138885615","432719396208","3269692046880","14359883150592","31436007144192","24982689005568","3108445175808"],["1","273","33138","2354121","108382050","3381856758","72635764284","1068204859029","10488403911888","65384283132000","236688434857728","423003242559744","268115731513344","25311625003008"],["1","294","38766","3023244","154812042","5462679852","135567962388","2371076212392","28822377334464","235921791490560","1229659528075776","3710465074520064","5468064651067392","2791778491072512","202493000024064"],["1","315","44835","3808350","214742367","8458008174","238367847330","4841551396260","70427864184993","719294099301288","4973126585285376","21910024872066240","55632567877622784","68190758524035072","28292068847112192","1594632375189504"],["1","336","51345","4718700","290562363","12640003992","399022726014","9241838373240","156900139064361","1929332508320448","16772131332579168","99004449637731456","372078217886964480","801445842630918144","823281820320264192","279779813592072192","12381851383824384"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["379","5713","80032","1080325","14354557","189853438","2511177679","33258000301","440953665088","5849978617585","77627638378537","1030136054227366","13669717178313811","181387989640645081","2406840013228174816"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","42","336"]],"recurrence":{"denominator":{"coeffs":[["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","84"],["1","42"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-36"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192189,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0fd836f092b28995"]},{"id":192190,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0fd836f092b28995"]},{"id":192191,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0fd836f092b28995"]}],"region_exact":{},"incoming_edges":[{"id":"e-0fd836f092b28995","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-ab60eb3242173f47","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"8c5a925d448ba1277acef790427d7bbac348d162e4937d53bac75116d5bf31c1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-636b5c0800d63e74","kind":"fingerprint_related","node_a_id":"n-44b1dbf5d009bf35","node_b_id":"n-ab60eb3242173f47","fingerprint":"f6-f34dcee26c4e5272","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f34dcee26c4e5272"},"other_id":"n-44b1dbf5d009bf35","other":{"id":"n-44b1dbf5d009bf35","name":"A convolution triangle of numbers obtained from A036224.","first_index":0,"fingerprint":"f6-f34dcee26c4e5272","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["336","42","1"],["4536","1113","63","1"],["54432","23184","2331","84","1"]],"card_rows":[["336","42","1"],["4536","1113","63","1"],["54432","23184","2331","84","1"],["598752","412272","65205","3990","105","1"]],"row_sums_preview":["379","5713","80032","1080325","14354557","189853438","2511177679","33258000301"],"regions":["[-36, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-44b1dbf5d009bf35","steps":[],"edges":[],"text":"A132166"},"headline":"A convolution triangle of numbers obtained from A036224.","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: 1,  42,   336;\n  1: 1,  63,  1113,    4536;\n  2: 1,  84,  2331,   23184,    54432;\n  3: 1, 105,  3990,   65205,   412272,     598752;\n  4: 1, 126,  6090,  139860,  1518048,    6531840,     6158592;\n  5: 1, 147,  8631,  256410,  4010769,   30912840,    94618368,     60046272;\n  6: 1, 168, 11613,  424116,  8723925,   99302112,   567362016,   1274921856,    560431872;\n  7: 1, 189, 15036,  652239, 16685487,  253790901,  2193126624,   9573741216,  16176101760,    5043886848;\n  8: 1, 210, 18900,  950040, 29117907,  557085438,  6534131760,  44153890560, 150628802688,  195030291456,   44019376128;\n  9: 1, 231, 23205, 1326780, 47438118, 1096591041, 16358573673, 152347155480, 822902685696, 2232875211840, 2250184914432, 374164697088;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = (n+3+(42*n+84)*x)*P(n-1,x) - (x+36*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 42*x + 336*x^2.","mathematica":"P[0, x_] := 1 + 42*x + 336*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 3 + (42*n + 84)*x)*P[n - 1, x] - (x + 36*x^2)*D[P[n - 1, x], x])/(n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 42*x + 336*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 3 + (42*n + 84)*x)*v[i - 1] - (x + 36*x^2)*deriv(v[i - 1], x))/(n + 3)); 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-44b1dbf5d009bf35","steps":["reverse"],"edges":["e-0fd836f092b28995"],"text":"A132166 \u2192 reverse"},"headline":"A132166 \u2192 reverse","chain":null}
