{"id":"n-22693abd431a71f2","revision":2,"first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(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"],["6","1"],["36","12","1"],["216","108","18","1"],["1296","864","216","24","1"],["7776","6480","2160","360","30","1"],["46656","46656","19440","4320","540","36","1"],["279936","326592","163296","45360","7560","756","42","1"],["1679616","2239488","1306368","435456","90720","12096","1008","48","1"],["10077696","15116544","10077696","3919104","979776","163296","18144","1296","54","1"],["60466176","100776960","75582720","33592320","9797760","1959552","272160","25920","1620","60","1"],["362797056","665127936","554273280","277136640","92378880","21555072","3592512","427680","35640","1980","66","1"],["2176782336","4353564672","3990767616","2217093120","831409920","221709312","43110144","6158592","641520","47520","2376","72","1"],["13060694016","28298170368","28298170368","17293326336","7205552640","2161665792","480370176","80061696","10007712","926640","61776","2808","78","1"],["78364164096","182849716224","198087192576","132058128384","60526642176","20175547392","5043886848","960740352","140107968","15567552","1297296","78624","3276","84","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","49","343","2401","16807","117649","823543","5764801","40353607","282475249","1977326743","13841287201","96889010407","678223072849"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038255","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158474,"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":false,"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-29094c360cdb0314"]},{"id":158475,"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-29094c360cdb0314","e-e8d1c828e3ebc417","e-efe1634c04c9236a"]},{"id":158476,"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-29094c360cdb0314","e-e8d1c828e3ebc417","e-efe1634c04c9236a"]},{"id":158477,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-29094c360cdb0314"]},{"id":158478,"position":4,"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-e8d1c828e3ebc417","e-efe1634c04c9236a"]},{"id":158479,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-d62fb68c5bc380ca"]},{"id":158480,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 6]","output_interval":"(-inf, 0]","statement":"an input root r <= 6 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 6] to roots in (-inf, 0]: an input root r <= 6 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-d62fb68c5bc380ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-efe1634c04c9236a","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-22693abd431a71f2","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a42620a3038512229792bdd9c81ce7e1f734bf9bf6ba78421af3cb096ae9c652","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-7b91a7a037fb3513","role":"sequence","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(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]"}]},{"id":"e-d62fb68c5bc380ca","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-22693abd431a71f2","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a9ac161f883efc0efd080efb00a1608ccd23a0c6c030f7755ca547367dfda5a6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 6]","output_interval":"(-inf, 0]","statement":"an input root r <= 6 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-e8d1c828e3ebc417","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-22693abd431a71f2","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"b83723caa0f4fbd797327fe48ffd0b0addcbfe7c016da4385de05e3cb8fc9f4d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-29094c360cdb0314","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-22693abd431a71f2","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"e75d164c2aae4fefc96384f344c3c401f35b229fc75306c971006e12e4d065ef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1b27bff9bafb5517","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-ade19c5257c8a726","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-4c464713c83c5f6b","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-6a4f4798749d2440","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-845ce04339675149","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-47ecd85a6f8f5bd0","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-b9cdb58d54bdc029","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-67dfd62e07739d13","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-cffbbf28e801bdbf","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-f211cb6153c37c6d","conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)."},{"id":"e-877f4cc430a42e49","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a5bec6dea8d577c1","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."},{"id":"e-877f4cc430a42e49","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-a5bec6dea8d577c1","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)."}],"used_by_total":7,"links":[{"id":"l-3e86ae850b54d3ab","kind":"fingerprint_related","node_a_id":"n-22693abd431a71f2","node_b_id":"n-ea9b82d727693fb6","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-ea9b82d727693fb6","other":{"id":"n-ea9b82d727693fb6","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"]],"card_rows":[["144","48","4"],["1728","864","144","8"],["20736","13824","3456","384","16"],["248832","207360","69120","11520","960","32"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":[],"edges":[],"text":"A038328"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","chain":null}},{"id":"l-b237552f6e549d24","kind":"fingerprint_related","node_a_id":"n-efce1b6c07b70e1a","node_b_id":"n-22693abd431a71f2","fingerprint":"f6-a5035a1c3b326aa3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5035a1c3b326aa3"},"other_id":"n-efce1b6c07b70e1a","other":{"id":"n-efce1b6c07b70e1a","name":"Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-a5035a1c3b326aa3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","6"],["1","12","36"]],"card_rows":[["1"],["1","6"],["1","12","36"],["1","18","108","216"]],"row_sums_preview":["1","7","49","343","2401","16807","117649","823543"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A013613"],"derivation":{"root":"n-efce1b6c07b70e1a","steps":[],"edges":[],"text":"A013613"},"headline":"A013613: Triangle of coefficients in expansion of (1+6x)^n.","chain":null}},{"id":"l-1c1a0b89ba857e37","kind":"oeis_match","node_a_id":"n-22693abd431a71f2","node_b_id":null,"fingerprint":null,"oeis_id":"A013613","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A013613","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a44d44bd498814a1","kind":"oeis_match","node_a_id":"n-22693abd431a71f2","node_b_id":null,"fingerprint":null,"oeis_id":"A000420","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000420","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b88aaf0734990cda","kind":"oeis_match","node_a_id":"n-22693abd431a71f2","node_b_id":null,"fingerprint":null,"oeis_id":"A038255","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":43},"in_catalog":true,"oeis_id":"A038255","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038255","relation":"identification"},{"oeis_id":"A013613","relation":"match_reversed_triangle"},{"oeis_id":"A000420","relation":"match_row_sums"},{"oeis_id":"A038255","relation":"match_triangle"}],"bfiles":[{"node_id":"n-22693abd431a71f2","oeis_id":"A000420","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":17808},{"node_id":"n-22693abd431a71f2","oeis_id":"A013613","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":21698},{"node_id":"n-22693abd431a71f2","oeis_id":"A038255","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":43,"oeis_terms_agreeing":43,"oeis_terms_available":43,"term_count":990,"raw_size":21698}],"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:        6,        1;\n  2:       36,       12,        1;\n  3:      216,      108,       18,       1;\n  4:     1296,      864,      216,      24,      1;\n  5:     7776,     6480,     2160,     360,     30,      1;\n  6:    46656,    46656,    19440,    4320,    540,     36,     1;\n  7:   279936,   326592,   163296,   45360,   7560,    756,    42,    1;\n  8:  1679616,  2239488,  1306368,  435456,  90720,  12096,  1008,   48,  1;\n  9: 10077696, 15116544, 10077696, 3919104, 979776, 163296, 18144, 1296, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+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[(6 + 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] = (6 + 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}},{"oeis_id":"A013613","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  6;\n  2: 1, 12,   36;\n  3: 1, 18,  108,   216;\n  4: 1, 24,  216,   864,   1296;\n  5: 1, 30,  360,  2160,   6480,   7776;\n  6: 1, 36,  540,  4320,  19440,  46656,   46656;\n  7: 1, 42,  756,  7560,  45360, 163296,  326592,   279936;\n  8: 1, 48, 1008, 12096,  90720, 435456, 1306368,  2239488,  1679616;\n  9: 1, 54, 1296, 18144, 163296, 979776, 3919104, 10077696, 15116544, 10077696;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+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[(6 + x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (6 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038255","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        6,        1;\n  2:       36,       12,        1;\n  3:      216,      108,       18,       1;\n  4:     1296,      864,      216,      24,      1;\n  5:     7776,     6480,     2160,     360,     30,      1;\n  6:    46656,    46656,    19440,    4320,    540,     36,     1;\n  7:   279936,   326592,   163296,   45360,   7560,    756,    42,    1;\n  8:  1679616,  2239488,  1306368,  435456,  90720,  12096,  1008,   48,  1;\n  9: 10077696, 15116544, 10077696, 3919104, 979776, 163296, 18144, 1296, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+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[(6 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (6 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-22693abd431a71f2","steps":[],"edges":[],"text":"A038255"},"headline":"A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","chain":null}