{"id":"n-b5c68d88e0751a6d","revision":0,"first_index":0,"fingerprint":"f6-d2d514e02d66ef29","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-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","7"],["1","14","49"],["1","21","147","343"],["1","28","294","1372","2401"],["1","35","490","3430","12005","16807"],["1","42","735","6860","36015","100842","117649"],["1","49","1029","12005","84035","352947","823543","823543"],["1","56","1372","19208","168070","941192","3294172","6588344","5764801"],["1","63","1764","28812","302526","2117682","9882516","29647548","51883209","40353607"],["1","70","2205","41160","504210","4235364","24706290","98825160","259416045","403536070","282475249"],["1","77","2695","56595","792330","7764834","54353838","271769190","951192165","2219448385","3107227739","1977326743"],["1","84","3234","75460","1188495","13311144","108707676","652246056","2853576495","8877793540","18643366434","23727920916","13841287201"],["1","91","3822","98098","1716715","21630609","201885684","1413199788","7419298887","28852829005","80787921214","154231485954","179936733613","96889010407"],["1","98","4459","124852","2403401","33647614","353299947","2826399576","17311697403","80787921214","282757724249","719746934452","1259557135291","1356446145698","678223072849"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","64","512","4096","32768","262144","2097152","16777216","134217728","1073741824","8589934592","68719476736","549755813888","4398046511104"],"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","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183796,"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-9f98b6c47fde2fcf"]},{"id":183797,"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-9f98b6c47fde2fcf"]},{"id":183798,"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-9f98b6c47fde2fcf"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f98b6c47fde2fcf","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-b5c68d88e0751a6d","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":"65097a4fa0cd837e17a3eb3a3fd58f92515bd6885d38662ef2bc02f1a0a63558","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-f211cb6153c37c6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-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-f63697729a40381a","kind":"candidate_equal","node_a_id":"n-91efc137acf0c8c1","node_b_id":"n-b5c68d88e0751a6d","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-91efc137acf0c8c1","other":{"id":"n-91efc137acf0c8c1","name":"mobius_x_over_x_plus_one of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","7"],["1","14","49"]],"card_rows":[["1"],["1","7"],["1","14","49"],["1","21","147","343"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["x \u21a6 x/(x+1)"],"edges":["e-521859cf025f0f48"],"text":"A013613 \u2192 x \u21a6 x/(x+1)"},"headline":"A013613 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-83141c07bc1fd7f0","kind":"fingerprint_related","node_a_id":"n-b5c68d88e0751a6d","node_b_id":"n-8852cf7cc7f2c858","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-8852cf7cc7f2c858","other":{"id":"n-8852cf7cc7f2c858","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","2"],["196","56","4"]],"card_rows":[["1"],["14","2"],["196","56","4"],["2744","1176","168","8"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-4bd827707ee8c2a0","e-f8ad5a4c47a4f61e"],"text":"A038327 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038327 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-95036129da3ba13b","kind":"fingerprint_related","node_a_id":"n-b5c68d88e0751a6d","node_b_id":"n-e14737d97ac22426","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-e14737d97ac22426","other":{"id":"n-e14737d97ac22426","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["56","8"],["1176","336","24"],["21952","9408","1344","64"]],"card_rows":[["56","8"],["1176","336","24"],["21952","9408","1344","64"],["384160","219520","47040","4480","160"]],"row_sums_preview":["64","1536","32768","655360","12582912","234881024","4294967296","77309411328"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["x \u21a6 x+1","D"],"edges":["e-02ed32e9293d9244","e-8738d0978947ed0c"],"text":"A038328 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038328 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-c7c7c5f5b9a913fe","kind":"fingerprint_related","node_a_id":"n-f211cb6153c37c6d","node_b_id":"n-b5c68d88e0751a6d","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-f211cb6153c37c6d","other":{"id":"n-f211cb6153c37c6d","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","1"],["49","14","1"]],"card_rows":[["1"],["7","1"],["49","14","1"],["343","147","21","1"]],"row_sums_preview":["1","8","64","512","4096","32768","262144","2097152"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A027466"],"derivation":{"root":"n-f211cb6153c37c6d","steps":[],"edges":[],"text":"A027466"},"headline":"A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","chain":null}},{"id":"l-ded16f1f5797970f","kind":"fingerprint_related","node_a_id":"n-b5c68d88e0751a6d","node_b_id":"n-eccac710ef17edb6","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-eccac710ef17edb6","other":{"id":"n-eccac710ef17edb6","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["4","56","196"],["8","168","1176","2744"],["16","448","4704","21952","38416"]],"card_rows":[["4","56","196"],["8","168","1176","2744"],["16","448","4704","21952","38416"],["32","1120","15680","109760","384160","537824"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["x \u21a6 x+1","reverse"],"edges":["e-02ed32e9293d9244","e-fa4d1cc4a872e75c"],"text":"A038328 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038328 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-e92c27909efa7efb","kind":"fingerprint_related","node_a_id":"n-1adce149c354da4e","node_b_id":"n-b5c68d88e0751a6d","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-1adce149c354da4e","other":{"id":"n-1adce149c354da4e","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["14","2"]],"card_rows":[["1"],["14","2"],["147","42","3"],["1372","588","84","4"]],"row_sums_preview":["0","1","16","192","2048","20480","196608","1835008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["D"],"edges":["e-e0e9d674222bcec2"],"text":"A027466 \u2192 D"},"headline":"A027466 \u2192 D","chain":null}},{"id":"l-f17bd5f3224a14b2","kind":"fingerprint_related","node_a_id":"n-ef03768afe1bf556","node_b_id":"n-b5c68d88e0751a6d","fingerprint":"f6-d2d514e02d66ef29","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d2d514e02d66ef29"},"other_id":"n-ef03768afe1bf556","other":{"id":"n-ef03768afe1bf556","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d2d514e02d66ef29","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","56","4"],["2744","1176","168","8"],["38416","21952","4704","448","16"]],"card_rows":[["196","56","4"],["2744","1176","168","8"],["38416","21952","4704","448","16"],["537824","384160","109760","15680","1120","32"]],"row_sums_preview":["256","4096","65536","1048576","16777216","268435456","4294967296","68719476736"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["x \u21a6 x+1"],"edges":["e-02ed32e9293d9244"],"text":"A038328 \u2192 x \u21a6 x+1"},"headline":"A038328 \u2192 x \u21a6 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,  7;\n  2: 1, 14,   49;\n  3: 1, 21,  147,   343;\n  4: 1, 28,  294,  1372,   2401;\n  5: 1, 35,  490,  3430,  12005,   16807;\n  6: 1, 42,  735,  6860,  36015,  100842,  117649;\n  7: 1, 49, 1029, 12005,  84035,  352947,  823543,   823543;\n  8: 1, 56, 1372, 19208, 168070,  941192, 3294172,  6588344,  5764801;\n  9: 1, 63, 1764, 28812, 302526, 2117682, 9882516, 29647548, 51883209, 40353607;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+7*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 + 7*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 + 7*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-f211cb6153c37c6d","steps":["reverse"],"edges":["e-9f98b6c47fde2fcf"],"text":"A027466 \u2192 reverse"},"headline":"A027466 \u2192 reverse","chain":null}
