{"id":"n-f8e5ffc520f5b05b","revision":0,"first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","name":"basis_map_A008277 of basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["256","420","100"],["4096","13480","7800","1000"],["65536","391440","415600","124000","10000"],["1048576","10832800","18876000","9860000","1800000","100000"],["16777216","292138560","787864000","642320000","199400000","24600000","1000000"],["268435456","7763374720","31284489600","37425360000","17589600000","3617600000","322000000","10000000"],["4294967296","204532097280","1203875372800","2034411456000","1359271200000","414657600000","60648000000","4080000000","100000000"],["68719476736","5360784202240","45384834393600","105621680704000","96463301760000","40960113600000","8755824000000","957360000000","50400000000","1000000000"],["1099511627776","140067584025600","1687461880192000","5312445656320000","6458161705600000","3668000515200000","1075043760000000","169891200000000","14412000000000","610000000000","10000000000"],["17592186044416","3652752300943360","62149303527168000","261247118992640000","414781512076800000","306669651059200000","118383330912000000","25361080800000000","3082464000000000","208780000000000","7260000000000","100000000000"],["281474976710656","95147481684971520","2273902449987481600","12638860508933120000","25840235866227200000","24388012090675200000","12063829659904000000","3364886257920000000","549527352000000000","52955320000000000","2929960000000000","85200000000000","1000000000000"],["4503599627370496","2476649273576366080","82811963016399052800","604126607910798336000","1573441813598054400000","1868011156646835200000","1160731175059456000000","410018514780160000000","86403488371200000000","11108537440000000000","869428560000000000","40034800000000000","988000000000000","10000000000000"],["72057594037927936","64437917109259223040","3005997161326129561600","28617943594060713984000","94154007640599029760000","139023154474671667200000","106895680870987008000000","46868904021688320000000","12394920031436800000000","2041539852352000000000","211939087360000000000","13738670400000000000","534716000000000000","11340000000000000","100000000000000"],["1152921504606846976","1676106420781119078400","108860276978833256448000","1346485376940054138880000","5558803863814152806400000","10117068271734320332800000","9514303290941729280000000","5099682554575065600000000","1658601363234816000000000","340352424663680000000000","45000332657280000000000","3850463344000000000000","210108080000000000000","7002800000000000000","129000000000000000","1000000000000000"],["18446744073709551616","43590296155355164508160","3935731035445808422912000","63026930109030822952960000","324757870142993098547200000","723314544572606670028800000","824257732828914628608000000","533715732602872934400000000","210222556416292992000000000","52663370646698240000000000","8623562834881280000000000","935161707916800000000000","67079332320000000000000","3123489600000000000000","90152000000000000000","1456000000000000000","10000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["776","26376","1006576","42517376","1964099776","98280859776","5285870693376","303624018136576","18527203812965376","1195560945854795776","81270654622308683776","5800044871056106725376","433292881535584965656576","33794902719106217140453376","2745567096080472703833931776"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["256","420","100"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197977,"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-bbddc24cbe0ad2ab"]},{"id":197978,"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-bbddc24cbe0ad2ab"]},{"id":197979,"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-bbddc24cbe0ad2ab"]}],"region_exact":{},"incoming_edges":[{"id":"e-bbddc24cbe0ad2ab","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-f8e5ffc520f5b05b","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9319e460b8d3594779e0441c350881f2a4b509542f261313916791e36ee91622","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-d524cbe046c3619e","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+5x)^n.","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-796f7ef8de95bf70","kind":"fingerprint_related","node_a_id":"n-034190847bef139d","node_b_id":"n-f8e5ffc520f5b05b","fingerprint":"f6-1cfe5e80b35ba55f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1cfe5e80b35ba55f"},"other_id":"n-034190847bef139d","other":{"id":"n-034190847bef139d","name":"basis_map_A008277 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","5"],["64","105","25"]],"card_rows":[["1"],["8","5"],["64","105","25"],["512","1685","975","125"]],"row_sums_preview":["1","13","194","3297","62911","1328668","30689059","767819217"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A008277"],"edges":["e-f7e2151b905e00c8"],"text":"A013622 \u2192 basis A008277"},"headline":"A013622 \u2192 basis A008277","chain":null}},{"id":"l-c0c7552f4e58bbb7","kind":"fingerprint_related","node_a_id":"n-f8e5ffc520f5b05b","node_b_id":"n-3226d385218bf3b4","fingerprint":"f6-1cfe5e80b35ba55f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1cfe5e80b35ba55f"},"other_id":"n-3226d385218bf3b4","other":{"id":"n-3226d385218bf3b4","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-1cfe5e80b35ba55f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["32","20"],["1024","1680","400"]],"card_rows":[["1"],["32","20"],["1024","1680","400"],["32768","107840","62400","8000"]],"row_sums_preview":["1","52","3104","211008","16105216","1360556032","125702385664","12579950051328"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013611","basis A008277"],"edges":["e-62a1b0a86f60f435","e-afd4691742acaf5a"],"text":"A038271 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038271 \u2192 basis A013611 \u2192 basis A008277","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:            256,              420,               100;\n  1:           4096,            13480,              7800,               1000;\n  2:          65536,           391440,            415600,             124000,              10000;\n  3:        1048576,         10832800,          18876000,            9860000,            1800000,             100000;\n  4:       16777216,        292138560,         787864000,          642320000,          199400000,           24600000,            1000000;\n  5:      268435456,       7763374720,       31284489600,        37425360000,        17589600000,         3617600000,          322000000,          10000000;\n  6:     4294967296,     204532097280,     1203875372800,      2034411456000,      1359271200000,       414657600000,        60648000000,        4080000000,        100000000;\n  7:    68719476736,    5360784202240,    45384834393600,    105621680704000,     96463301760000,     40960113600000,      8755824000000,      957360000000,      50400000000,      1000000000;\n  8:  1099511627776,  140067584025600,  1687461880192000,   5312445656320000,   6458161705600000,   3668000515200000,   1075043760000000,   169891200000000,   14412000000000,    610000000000,   10000000000;\n  9: 17592186044416, 3652752300943360, 62149303527168000, 261247118992640000, 414781512076800000, 306669651059200000, 118383330912000000, 25361080800000000, 3082464000000000, 208780000000000, 7260000000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+10*x)*P(n-1,x) + 10*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 420*x + 100*x^2.","mathematica":"P[0, x_] := 256 + 420*x + 100*x^2;\nP[n_, x_] := P[n, x] = Expand[(16 + 10*x)*P[n - 1, x] + 10*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 256 + 420*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 10*x)*v[i - 1] + 10*x*deriv(v[i - 1], x)); 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-bbfc5a54ff13a785","steps":["basis A013609","basis A008277"],"edges":["e-31086658f6c15e80","e-bbddc24cbe0ad2ab"],"text":"A013612 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A013612 \u2192 basis A013609 \u2192 basis A008277","chain":null}