{"id":"n-6608547fbf1f3999","revision":0,"first_index":0,"fingerprint":"f6-a8581dd4789601b6","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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":true,"rows":[["1"],["1","12"],["1","24","144"],["1","36","432","1728"],["1","48","864","6912","20736"],["1","60","1440","17280","103680","248832"],["1","72","2160","34560","311040","1492992","2985984"],["1","84","3024","60480","725760","5225472","20901888","35831808"],["1","96","4032","96768","1451520","13934592","83607552","286654464","429981696"],["1","108","5184","145152","2612736","31352832","250822656","1289945088","3869835264","5159780352"],["1","120","6480","207360","4354560","62705664","627056640","4299816960","19349176320","51597803520","61917364224"],["1","132","7920","285120","6842880","114960384","1379524608","11824496640","70946979840","283787919360","681091006464","743008370688"],["1","144","9504","380160","10264320","197074944","2759049216","28378791936","212840939520","1135151677440","4086546038784","8916100448256","8916100448256"],["1","156","11232","494208","14826240","320246784","5123948544","61487382528","553386442752","3689242951680","17708366168064","57954652913664","115909305827328","106993205379072"],["1","168","13104","628992","20756736","498161664","8966909952","122974765056","1291235033088","10329880264704","61979281588224","270455046930432","811365140791296","1497904875307008","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","169","2197","28561","371293","4826809","62748517","815730721","10604499373","137858491849","1792160394037","23298085122481","302875106592253","3937376385699289"],"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":[["1","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188033,"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-d8646f4cd423cfd3"]},{"id":188034,"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-d8646f4cd423cfd3"]},{"id":188035,"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-d8646f4cd423cfd3"]}],"region_exact":{},"incoming_edges":[{"id":"e-d8646f4cd423cfd3","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-6608547fbf1f3999","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":"12d8007501995950633bfbcd24ae14302e5cf2263ba7b26098a938a59cb1fe29","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-81e6fb4f9f226d77","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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-42e01fb6548c7fcc","kind":"fingerprint_related","node_a_id":"n-81e6fb4f9f226d77","node_b_id":"n-6608547fbf1f3999","fingerprint":"f6-a8581dd4789601b6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8581dd4789601b6"},"other_id":"n-81e6fb4f9f226d77","other":{"id":"n-81e6fb4f9f226d77","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8581dd4789601b6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","1"],["144","24","1"]],"card_rows":[["1"],["12","1"],["144","24","1"],["1728","432","36","1"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038327"],"derivation":{"root":"n-81e6fb4f9f226d77","steps":[],"edges":[],"text":"A038327"},"headline":"A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","chain":null}},{"id":"l-77f14ba3ec3125d7","kind":"fingerprint_related","node_a_id":"n-cae4d1df47d65e38","node_b_id":"n-6608547fbf1f3999","fingerprint":"f6-a8581dd4789601b6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8581dd4789601b6"},"other_id":"n-cae4d1df47d65e38","other":{"id":"n-cae4d1df47d65e38","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-a8581dd4789601b6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["24","2"]],"card_rows":[["1"],["24","2"],["432","72","3"],["6912","1728","144","4"]],"row_sums_preview":["0","1","26","507","8788","142805","2227758","33787663"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["D"],"edges":["e-b5bcd2b1fd1856df"],"text":"A038327 \u2192 D"},"headline":"A038327 \u2192 D","chain":null}},{"id":"l-fe4e5bfca2bb1dc0","kind":"fingerprint_related","node_a_id":"n-6608547fbf1f3999","node_b_id":"n-02a5702f23ec3d86","fingerprint":"f6-a8581dd4789601b6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a8581dd4789601b6"},"other_id":"n-02a5702f23ec3d86","other":{"id":"n-02a5702f23ec3d86","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-a8581dd4789601b6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","24","144"],["1","36","432","1728"],["1","48","864","6912","20736"]],"card_rows":[["1","24","144"],["1","36","432","1728"],["1","48","864","6912","20736"],["1","60","1440","17280","103680","248832"]],"row_sums_preview":["169","2197","28561","371293","4826809","62748517","815730721","10604499373"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["x \u21a6 x/(x+1)"],"edges":["e-d67b1add67326e2e"],"text":"A013618 \u2192 x \u21a6 x/(x+1)"},"headline":"A013618 \u2192 x \u21a6 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: 1;\n  1: 1,  12;\n  2: 1,  24,  144;\n  3: 1,  36,  432,   1728;\n  4: 1,  48,  864,   6912,   20736;\n  5: 1,  60, 1440,  17280,  103680,   248832;\n  6: 1,  72, 2160,  34560,  311040,  1492992,   2985984;\n  7: 1,  84, 3024,  60480,  725760,  5225472,  20901888,   35831808;\n  8: 1,  96, 4032,  96768, 1451520, 13934592,  83607552,  286654464,  429981696;\n  9: 1, 108, 5184, 145152, 2612736, 31352832, 250822656, 1289945088, 3869835264, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 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-81e6fb4f9f226d77","steps":["reverse"],"edges":["e-d8646f4cd423cfd3"],"text":"A038327 \u2192 reverse"},"headline":"A038327 \u2192 reverse","chain":null}
