{"id":"n-e1ce1a92fab8fc7c","revision":0,"first_index":0,"fingerprint":"f6-d935ec63feee39a0","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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"],["16","15"],["256","480","225"],["4096","11520","10800","3375"],["65536","245760","345600","216000","50625"],["1048576","4915200","9216000","8640000","4050000","759375"],["16777216","94371840","221184000","276480000","194400000","72900000","11390625"],["268435456","1761607680","4954521600","7741440000","7257600000","4082400000","1275750000","170859375"],["4294967296","32212254720","105696460800","198180864000","232243200000","174182400000","81648000000","21870000000","2562890625"],["68719476736","579820584960","2174327193600","4756340736000","6688604160000","6270566400000","3919104000000","1574640000000","369056250000","38443359375"],["1099511627776","10307921510400","43486543872000","108716359680000","178362777600000","200658124800000","156764160000000","83980800000000","29524500000000","6150937500000","576650390625"],["17592186044416","181419418583040","850403524608000","2391759912960000","4484549836800000","5885971660800000","5518098432000000","3695155200000000","1732104000000000","541282500000000","101490468750000","8649755859375"],["281474976710656","3166593487994880","16327747672473600","51024211476480000","107629196083200000","161443794124800000","176579149824000000","141893959680000000","83140992000000000","34642080000000000","9743085000000000","1660753125000000","129746337890625"],["4503599627370496","54887620458577920","308742865079500800","1061303598710784000","2487430309478400000","4197538647244800000","5246923309056000000","4918990602240000000","3458665267200000000","1801388160000000000","675520560000000000","172718325000000000","26987238281250000","1946195068359375"],["72057594037927936","945755921747804160","5763200148150681600","21612000555565056000","55718438932316160000","104472072998092800000","146913852653568000000","157407699271680000000","129123503308800000000","80702189568000000000","37829151360000000000","12896301600000000000","3022570687500000000","435947695312500000","29192926025390625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","31","961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671","819628286980801","25408476896404831","787662783788549761","24417546297445042591","756943935220796320321"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203534,"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-e5d188fb163a2e02"]},{"id":203535,"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-e5d188fb163a2e02"]},{"id":203536,"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-e5d188fb163a2e02"]}],"region_exact":{},"incoming_edges":[{"id":"e-e5d188fb163a2e02","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-e1ce1a92fab8fc7c","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":"527daa2e755c30064ad855d90f47b992e3aeb1ec87da74d21a61637213d3ca95","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-0195873a37fb074a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^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-30eda435ea309e18","kind":"fingerprint_related","node_a_id":"n-a875e5d4789de94f","node_b_id":"n-e1ce1a92fab8fc7c","fingerprint":"f6-d935ec63feee39a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d935ec63feee39a0"},"other_id":"n-a875e5d4789de94f","other":{"id":"n-a875e5d4789de94f","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-d935ec63feee39a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","480","225"],["4096","11520","10800","3375"],["65536","245760","345600","216000","50625"]],"card_rows":[["256","480","225"],["4096","11520","10800","3375"],["65536","245760","345600","216000","50625"],["1048576","4915200","9216000","8640000","4050000","759375"]],"row_sums_preview":["961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013610"],"edges":["e-fd6222a072627022"],"text":"A038319 \u2192 basis A013610"},"headline":"A038319 \u2192 basis A013610","chain":null}},{"id":"l-31d3c32631fd98db","kind":"fingerprint_related","node_a_id":"n-98095ae5bcb61b1c","node_b_id":"n-e1ce1a92fab8fc7c","fingerprint":"f6-d935ec63feee39a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d935ec63feee39a0"},"other_id":"n-98095ae5bcb61b1c","other":{"id":"n-98095ae5bcb61b1c","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-d935ec63feee39a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["480","450"],["11520","21600","10125"],["245760","691200","648000","202500"]],"card_rows":[["480","450"],["11520","21600","10125"],["245760","691200","648000","202500"],["4915200","18432000","25920000","16200000","3796875"]],"row_sums_preview":["930","43245","1787460","69264075","2576623590","93187886505","3301513693320","115140290054535"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013610","D"],"edges":["e-fd6222a072627022","e-1f5a2970729c2967"],"text":"A038319 \u2192 basis A013610 \u2192 D"},"headline":"A038319 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-95803097ded428c2","kind":"fingerprint_related","node_a_id":"n-a8ec826a617e4dc9","node_b_id":"n-e1ce1a92fab8fc7c","fingerprint":"f6-d935ec63feee39a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d935ec63feee39a0"},"other_id":"n-a8ec826a617e4dc9","other":{"id":"n-a8ec826a617e4dc9","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-d935ec63feee39a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","480","256"],["3375","10800","11520","4096"],["50625","216000","345600","245760","65536"]],"card_rows":[["225","480","256"],["3375","10800","11520","4096"],["50625","216000","345600","245760","65536"],["759375","4050000","8640000","9216000","4915200","1048576"]],"row_sums_preview":["961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013610","reverse"],"edges":["e-fd6222a072627022","e-50bab57899c905ca"],"text":"A038319 \u2192 basis A013610 \u2192 reverse"},"headline":"A038319 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-a6d5db5eadb3ea9b","kind":"fingerprint_related","node_a_id":"n-2e9b5735d80321d0","node_b_id":"n-e1ce1a92fab8fc7c","fingerprint":"f6-d935ec63feee39a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d935ec63feee39a0"},"other_id":"n-2e9b5735d80321d0","other":{"id":"n-2e9b5735d80321d0","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-d935ec63feee39a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","1920","900"],["32768","92160","86400","27000"],["1048576","3932160","5529600","3456000","810000"]],"card_rows":[["1024","1920","900"],["32768","92160","86400","27000"],["1048576","3932160","5529600","3456000","810000"],["33554432","157286400","294912000","276480000","129600000","24300000"]],"row_sums_preview":["3844","238328","14776336","916132832","56800235584","3521614606208","218340105584896","13537086546263552"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-0c3feb8686271cde","e-3a548b292216a330"],"text":"A038336 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038336 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-c989738217a9810d","kind":"fingerprint_related","node_a_id":"n-748ef5bd7f0b1a24","node_b_id":"n-e1ce1a92fab8fc7c","fingerprint":"f6-d935ec63feee39a0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d935ec63feee39a0"},"other_id":"n-748ef5bd7f0b1a24","other":{"id":"n-748ef5bd7f0b1a24","name":"mobius_x_over_x_plus_one of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-d935ec63feee39a0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","16"],["225","480","256"]],"card_rows":[["1"],["15","16"],["225","480","256"],["3375","10800","11520","4096"]],"row_sums_preview":["1","31","961","29791","923521","28629151","887503681","27512614111"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["x \u21a6 x/(x+1)"],"edges":["e-f043aa3518173576"],"text":"A027467 \u2192 x \u21a6 x/(x+1)"},"headline":"A027467 \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:          16,           15;\n  2:         256,          480,           225;\n  3:        4096,        11520,         10800,          3375;\n  4:       65536,       245760,        345600,        216000,         50625;\n  5:     1048576,      4915200,       9216000,       8640000,       4050000,        759375;\n  6:    16777216,     94371840,     221184000,     276480000,     194400000,      72900000,      11390625;\n  7:   268435456,   1761607680,    4954521600,    7741440000,    7257600000,    4082400000,    1275750000,     170859375;\n  8:  4294967296,  32212254720,  105696460800,  198180864000,  232243200000,  174182400000,   81648000000,   21870000000,   2562890625;\n  9: 68719476736, 579820584960, 2174327193600, 4756340736000, 6688604160000, 6270566400000, 3919104000000, 1574640000000, 369056250000, 38443359375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+15*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[(16 + 15*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] = (16 + 15*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-e6ebe74518d42cbe","steps":["basis A007318","basis A013610"],"edges":["e-1194f7dc6f45a71b","e-e5d188fb163a2e02"],"text":"A038259 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A038259 \u2192 basis A007318 \u2192 basis A013610","chain":null}
