{"id":"n-e9c94c7bcfd7f102","revision":0,"first_index":0,"fingerprint":"f6-dd0a803b420669cd","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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"],["32","60"],["1024","3840","3600"],["32768","184320","345600","216000"],["1048576","7864320","22118400","27648000","12960000"],["33554432","314572800","1179648000","2211840000","2073600000","777600000"],["1073741824","12079595520","56623104000","141557760000","199065600000","149299200000","46656000000"],["34359738368","450971566080","2536715059200","7927234560000","14863564800000","16721510400000","10450944000000","2799360000000"],["1099511627776","16492674416640","108233175859200","405874409472000","951268147200000","1426902220800000","1337720832000000","716636160000000","167961600000000"],["35184372088832","593736278999040","4453022092492800","19481971654656000","54793045278720000","102736959897600000","128421199872000000","103195607040000000","48372940800000000","10077696000000000"],["1125899906842624","21110623253299200","178120883699712000","890604418498560000","2922295748198400000","6575165433446400000","10273695989760000000","11007531417600000000","7739670528000000000","3224862720000000000","604661760000000000"],["36028797018963968","743093938516131840","6966505673588736000","39186594413936640000","146949729052262400000","385743038762188800000","723268197679104000000","968662764748800000000","908121341952000000000","567575838720000000000","212840939520000000000","36279705600000000000"],["1152921504606846976","25940733853654056960","267513817865807462400","1671961361661296640000","7053586994508595200000","21160760983525785600000","46289164651462656000000","74393300332707840000000","87179648827392000000000","72649707356160000000000","40865460387840000000000","13931406950400000000000","2176782336000000000000"],["36893488147419103232","899278773593340641280","10116886202925082214400","69553592645109940224000","326032465523952844800000","1100359571143340851200000","2750898927858352128000000","5157935489734410240000000","7253346782439014400000000","7555569565040640000000000","5666677173780480000000000","2897732645683200000000000","905541451776000000000000","130606940160000000000000"],["1180591620717411303424","30990530043832046714880","377697084909203069337600","2832728136819023020032000","14606254455473087447040000","54773454208024077926400000","154050339960067719168000000","330107871343002255360000000","541583226422113075200000000","676979033027641344000000000","634667843463413760000000000","432728075088691200000000000","202841285197824000000000000","58511909191680000000000000","7836416409600000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","92","8464","778688","71639296","6590815232","606355001344","55784660123648","5132188731375616","472161363286556672","43438845422363213824","3996373778857415671808","367666387654882241806336","33825307664249166246182912","3111928305110923294648827904"],"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":[["32","60"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203821,"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-15c4c7a0aa609a8a"]},{"id":203822,"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-15c4c7a0aa609a8a"]},{"id":203823,"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-15c4c7a0aa609a8a"]}],"region_exact":{},"incoming_edges":[{"id":"e-15c4c7a0aa609a8a","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-e9c94c7bcfd7f102","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":"5e20de4eb593cae6e4e62cafee56dfef6cb1921184066b859cd5cb900f7487bb","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-384c17a53af3442c","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-4ebafc1e42a7647f","kind":"fingerprint_related","node_a_id":"n-dd3479747c695235","node_b_id":"n-e9c94c7bcfd7f102","fingerprint":"f6-dd0a803b420669cd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd0a803b420669cd"},"other_id":"n-dd3479747c695235","other":{"id":"n-dd3479747c695235","name":"reversal of basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-dd0a803b420669cd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","8"],["225","240","64"]],"card_rows":[["1"],["15","8"],["225","240","64"],["3375","5400","2880","512"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013610","reverse"],"edges":["e-c78a98c6ca91439c","e-0113e8b59838077b"],"text":"A013622 \u2192 basis A013610 \u2192 reverse"},"headline":"A013622 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-b76d4437dac13101","kind":"fingerprint_related","node_a_id":"n-82784edb4f691ed1","node_b_id":"n-e9c94c7bcfd7f102","fingerprint":"f6-dd0a803b420669cd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd0a803b420669cd"},"other_id":"n-82784edb4f691ed1","other":{"id":"n-82784edb4f691ed1","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-dd0a803b420669cd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["15"],["240","450"]],"card_rows":[["15"],["240","450"],["2880","10800","10125"],["30720","172800","324000","202500"]],"row_sums_preview":["0","15","690","23805","730020","20988075","579270870","15543768345"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013610","D"],"edges":["e-c78a98c6ca91439c","e-05c8984faeb21c94"],"text":"A013622 \u2192 basis A013610 \u2192 D"},"headline":"A013622 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-d2e020c66cdc991f","kind":"fingerprint_related","node_a_id":"n-a6cd3f3657ee08fd","node_b_id":"n-e9c94c7bcfd7f102","fingerprint":"f6-dd0a803b420669cd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd0a803b420669cd"},"other_id":"n-a6cd3f3657ee08fd","other":{"id":"n-a6cd3f3657ee08fd","name":"basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-dd0a803b420669cd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","15"],["64","240","225"]],"card_rows":[["1"],["8","15"],["64","240","225"],["512","2880","5400","3375"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013610"],"edges":["e-c78a98c6ca91439c"],"text":"A013622 \u2192 basis A013610"},"headline":"A013622 \u2192 basis A013610","chain":null}},{"id":"l-fbd8785be2c06615","kind":"fingerprint_related","node_a_id":"n-69d39bd7c8ec9d15","node_b_id":"n-e9c94c7bcfd7f102","fingerprint":"f6-dd0a803b420669cd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd0a803b420669cd"},"other_id":"n-69d39bd7c8ec9d15","other":{"id":"n-69d39bd7c8ec9d15","name":"basis_map_A013610 of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-dd0a803b420669cd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","960","900"],["4096","23040","43200","27000"],["65536","491520","1382400","1728000","810000"]],"card_rows":[["256","960","900"],["4096","23040","43200","27000"],["65536","491520","1382400","1728000","810000"],["1048576","9830400","36864000","69120000","64800000","24300000"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013609","basis A013610"],"edges":["e-31086658f6c15e80","e-1fb83deddcb4b215"],"text":"A013612 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A013612 \u2192 basis A013609 \u2192 basis A013610","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:             32,              60;\n  2:           1024,            3840,             3600;\n  3:          32768,          184320,           345600,            216000;\n  4:        1048576,         7864320,         22118400,          27648000,          12960000;\n  5:       33554432,       314572800,       1179648000,        2211840000,        2073600000,          777600000;\n  6:     1073741824,     12079595520,      56623104000,      141557760000,      199065600000,       149299200000,        46656000000;\n  7:    34359738368,    450971566080,    2536715059200,     7927234560000,    14863564800000,     16721510400000,     10450944000000,      2799360000000;\n  8:  1099511627776,  16492674416640,  108233175859200,   405874409472000,   951268147200000,   1426902220800000,   1337720832000000,    716636160000000,   167961600000000;\n  9: 35184372088832, 593736278999040, 4453022092492800, 19481971654656000, 54793045278720000, 102736959897600000, 128421199872000000, 103195607040000000, 48372940800000000, 10077696000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (32+60*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[(32 + 60*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] = (32 + 60*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-a6a633b989da1879","steps":["basis A013611","basis A013610"],"edges":["e-62a1b0a86f60f435","e-15c4c7a0aa609a8a"],"text":"A038271 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038271 \u2192 basis A013611 \u2192 basis A013610","chain":null}
