{"id":"n-9797801130dd5d88","revision":0,"first_index":0,"fingerprint":"f6-537eddf8e19d10b5","name":"reversal of Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["9","15","1"],["27","108","63","1"],["81","594","945","255","1"],["243","2835","8775","7380","1023","1"],["729","12393","63180","109890","54729","4095","1"],["2187","51030","387828","1151010","1263087","395388","16383","1"],["6561","201204","2133054","9658278","18752391","13817034","2816865","65535","1"],["19683","767637","10825650","69502860","210789621","285232185","146620935","19914660","262143","1"],["59049","2854035","51667875","446672880","1952923203","4228330491","4147881210","1525953330","140189049","1048575","1"],["177147","10392624","234916605","2631715515","15685572969","49790532330","80096931486","58500315720","15680100447","984469068","4194303","1"],["531441","37200870","1026921159","14472811485","112849606782","494454202308","1186310908728","1457051850936","807544405701","159754411674","6903866385","16777215","1"],["1594323","131265927","4345593057","75252990384","743787541926","4304602776474","14436925176960","26911062818640","25735462832079","10977340509135","1618255715895","48377396340","67108863","1"],["4782969","457570701","17893618470","373509135090","4564205327682","33739859503350","150925844942730","398345542349040","588516582050397","444699426840669","147560193766440","16327689347970","338843100969","268435455","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","25","199","1876","20257","245017","3266914","47450923","743935375","12497579698","223619318215","4240423494685","84855613320004","1785410320771933"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","-2"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185058,"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-4596ce48d1ceaad9"]},{"id":185059,"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-4596ce48d1ceaad9"]},{"id":185060,"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-4596ce48d1ceaad9"]}],"region_exact":{},"incoming_edges":[{"id":"e-4596ce48d1ceaad9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-9797801130dd5d88","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":"d411fd883e2723ca4ba9aa67498020161cc5da3c87a1dfddbaa4f41455723d35","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-315041bb871f134e","role":"sequence","name":"Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","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-4f57d54c3bcc2a06","kind":"fingerprint_related","node_a_id":"n-315041bb871f134e","node_b_id":"n-9797801130dd5d88","fingerprint":"f6-537eddf8e19d10b5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-537eddf8e19d10b5"},"other_id":"n-315041bb871f134e","other":{"id":"n-315041bb871f134e","name":"Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","first_index":0,"fingerprint":"f6-537eddf8e19d10b5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","15","9"]],"card_rows":[["1"],["1","3"],["1","15","9"],["1","63","108","27"]],"row_sums_preview":["1","4","25","199","1876","20257","245017","3266914"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A282629"],"derivation":{"root":"n-315041bb871f134e","steps":[],"edges":[],"text":"A282629"},"headline":"A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].","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:     1;\n  1:     3,      1;\n  2:     9,     15,        1;\n  3:    27,    108,       63,        1;\n  4:    81,    594,      945,      255,         1;\n  5:   243,   2835,     8775,     7380,      1023,         1;\n  6:   729,  12393,    63180,   109890,     54729,      4095,         1;\n  7:  2187,  51030,   387828,  1151010,   1263087,    395388,     16383,        1;\n  8:  6561, 201204,  2133054,  9658278,  18752391,  13817034,   2816865,    65535,      1;\n  9: 19683, 767637, 10825650, 69502860, 210789621, 285232185, 146620935, 19914660, 262143, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(3*n-2)*x)*P(n-1,x) - 3*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (3*n - 2)*x)*P[n - 1, x] - 3*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (3*n - 2)*x)*v[i - 1] - 3*x^2*deriv(v[i - 1], x)); 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-315041bb871f134e","steps":["reverse"],"edges":["e-4596ce48d1ceaad9"],"text":"A282629 \u2192 reverse"},"headline":"A282629 \u2192 reverse","chain":null}