{"id":"n-7da3df1306d64907","revision":0,"first_index":0,"fingerprint":"f6-8ceec15cbd981f14","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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"],["41","20"],["1281","1640","400"],["35721","76860","49200","8000"],["935361","2857680","3074400","1312000","160000"],["23532201","93536100","142884000","102480000","32800000","3200000"],["575858241","2823864120","5612166000","5715360000","3074400000","787200000","64000000"],["13808345481","80620153740","197670488400","261901080000","200037600000","86083200000","18368000000","1280000000"],["325997025921","2209335276960","6449612299200","10542426048000","10476043200000","6401203200000","2295552000000","419840000000","25600000000"],["7602394731561","58679464665780","198840174926400","386976737952000","474409172160000","377137555200000","192036096000000","59028480000000","9446400000000","512000000000"],["175535890294401","1520478946312200","5867946466578000","13256011661760000","19348836897600000","18976366886400000","12571251840000000","5486745600000000","1475712000000000","209920000000000","10240000000000"],["4019851315746441","38617895864768220","167252684094342000","430316074215720000","729080641396800000","851348823494400000","695800119168000000","395096486400000000","150885504000000000","36072960000000000","4618240000000000","204800000000000"],["91422427641519681","964764315779145840","4634147503772186400","13380214727547360000","25818964452943200000","34995870787046400000","34053952939776000000","23856004085760000000","11852894592000000000","4023613440000000000","865751040000000000","100761600000000000","4096000000000000"],["2066987530699646121","23769831186795117060","125419361051288959200","401626116993589488000","869713957290578400000","1342586151553046400000","1516487734105344000000","1264861109191680000000","775320132787200000000","342416954880000000000","104613949440000000000","20463206400000000000","2183168000000000000","81920000000000000"],["46496185699474957761","578756508595900913880","3327776366151316388400","11705807031453636192000","28113828189551264160000","48703981608272390400000","62654020405808832000000","60659509364213760000000","44270138821708800000000","24121070797824000000000","9587674736640000000000","2662900531200000000000","477474816000000000000","47022080000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","61","3321","169781","8339441","398432301","18652848361","859768867621","39145609050081","1764668475635741","78899046188944601","3503314079949776661","154642457912265811921","6791610580410222010381","296958095843119375012041"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","20"],["20","0"]]},"terms":[{"coeff":{"coeffs":[["441","840","400"],["420","400","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193148,"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-a1e386077dd5a3c1"]},{"id":193149,"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-a1e386077dd5a3c1"]},{"id":193150,"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-a1e386077dd5a3c1"]},{"id":193151,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a1e386077dd5a3c1"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1e386077dd5a3c1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7da3df1306d64907","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a039794e7399cbb3174b169593ea8f70e8799cafaff260dc1815e74a28286723","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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":[],"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:            41,             20;\n  2:          1281,           1640,             400;\n  3:         35721,          76860,           49200,            8000;\n  4:        935361,        2857680,         3074400,         1312000,          160000;\n  5:      23532201,       93536100,       142884000,       102480000,        32800000,         3200000;\n  6:     575858241,     2823864120,      5612166000,      5715360000,      3074400000,       787200000,        64000000;\n  7:   13808345481,    80620153740,    197670488400,    261901080000,    200037600000,     86083200000,     18368000000,     1280000000;\n  8:  325997025921,  2209335276960,   6449612299200,  10542426048000,  10476043200000,   6401203200000,   2295552000000,   419840000000,   25600000000;\n  9: 7602394731561, 58679464665780, 198840174926400, 386976737952000, 474409172160000, 377137555200000, 192036096000000, 59028480000000, 9446400000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy (20*n+1+20*x)*P(n,x) = (420*n+441+(400*n+840)*x+400*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((420*n + 441 + (400*n + 840)*x + 400*x^2)*P[n - 1, x])/(20*n + 1 + 20*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] = ((420*n + 441 + (400*n + 840)*x + 400*x^2)*v[i - 1])/(20*n + 1 + 20*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-e1d1088f7d1ff894","steps":["basis A013609","D+1"],"edges":["e-25d25e50279a57da","e-a1e386077dd5a3c1"],"text":"A038324 \u2192 basis A013609 \u2192 D+1"},"headline":"A038324 \u2192 basis A013609 \u2192 D+1","chain":null}