{"id":"n-aee3f5e748e8b8e4","revision":0,"first_index":0,"fingerprint":"f6-29d09fbfedb73813","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","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":true,"rows":[["385","552","144"],["5687","13860","9936","1728"],["78529","272976","332640","158976","20736"],["1039511","4711740","8189280","6652800","2384640","248832"],["13367233","74844792","169622640","196542720","119750400","34338816","2985984"],["168298295","1122847572","3143481264","4749433920","4127397120","2011806720","480743424","35831808"],["2085127297","16156636320","53896683456","100591400448","113986414080","79246024704","32188907520","6593052672","429981696"],["25508706839","225193748076","872458361280","1940280604416","2715967812096","2462106544128","1426428444672","496628858880","89006211072","5159780352"],["308891147521","3061044820680","13511624884560","34898334451200","58208418132480","65183227490304","49242130882560","24453059051520","7449432883200","1186749480960","61917364224"],["3709051717943","40773631472772","202028958164880","594511494920640","1151645036889600","1536702238697472","1434031004786688","928565896642560","403475474350080","109258348953600","15665093148672","743008370688"],["44223308944705","534103447383792","2935701466039584","9697389991914240","21402413817143040","33167377062420480","36880853728739328","29500066384183296","16714186139566080","6455607589601280","1573320224931840","205070310309888","8916100448256"],["524117538912407","6898836195373980","41660068895935776","152656476234058368","378198209684655360","667755311094862848","862351803622932480","821916168811905024","575251294491574272","289712559752478720","100707478397779968","22312541371760640","2665914034028544","106993205379072"],["6179565473763649","88051746537284376","579502240411414320","2332963858172403456","6411572001830451456","12707459845404420096","18697148710656159744","20696443286950379520","17260239545050005504","10738024163842719744","4867171003841642496","1538077851893366784","312375579204648960","34451812132061184","1283918464548864"],["72532218214399031","1112321785277456820","7924657188355593840","34770134424684859200","104983373617758155520","230816592065896252416","381223795362132602880","480783823988301250560","465669973956383539200","345204790901000110080","193284434949168955392","79644616426499604480","23071167778400501760","4325200327448985600","442951870269358080","15407021574586368"],["847981378391377153","13926185897164613952","106782891386635854720","507178060054758005760","1668966452384873241600","4031361546921913171968","7386130946108680077312","10456424101361351393280","11538811775719230013440","9934292777736182169600","6627931985299202113536","3373691955476403585024","1274313862823993671680","340743401034838179840","59317033062157516800","5669783939447783424","184884258895036416"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1081","31211","863857","23226803","611452585","15839840123","405174228193","10258739071811","257564830589209","6421109238115595","159119229571625809","3922717773331637459","96270945129865270153","2353345773881366211227","57326575624844117805505"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["385","552","144"]],"recurrence":{"denominator":{"coeffs":[["23","12"],["12","0"]]},"terms":[{"coeff":{"coeffs":[["385","552","144"],["132","144","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193904,"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-92deea8cc97bba0a"]},{"id":193905,"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-92deea8cc97bba0a"]},{"id":193906,"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-92deea8cc97bba0a"]},{"id":193907,"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-92deea8cc97bba0a"]}],"region_exact":{},"incoming_edges":[{"id":"e-92deea8cc97bba0a","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-aee3f5e748e8b8e4","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":"bbcf4e092f4354733cee200b58b3ea46b915b8bbfb79885603b4b81145774ede","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-f55a7ba0b32633bc","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-94a75420fb6a11ef","kind":"fingerprint_related","node_a_id":"n-aee3f5e748e8b8e4","node_b_id":"n-610f511ce0971e21","fingerprint":"f6-29d09fbfedb73813","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-29d09fbfedb73813"},"other_id":"n-610f511ce0971e21","other":{"id":"n-610f511ce0971e21","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-29d09fbfedb73813","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1540","2208","576"],["45496","110880","79488","13824"],["1256464","4367616","5322240","2543616","331776"]],"card_rows":[["1540","2208","576"],["45496","110880","79488","13824"],["1256464","4367616","5322240","2543616","331776"],["33264352","150775680","262056960","212889600","76308480","7962624"]],"row_sums_preview":["4324","249688","13821712","743257696","39132965440","2027499535744","103724602417408","5252474404767232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013609","D+1"],"edges":["e-61350e73fec19600","e-acf8a366e3346b88"],"text":"A038314 \u2192 basis A013609 \u2192 D+1"},"headline":"A038314 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-ad2a6811a7eba072","kind":"fingerprint_related","node_a_id":"n-728fca7147e339bb","node_b_id":"n-aee3f5e748e8b8e4","fingerprint":"f6-29d09fbfedb73813","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-29d09fbfedb73813"},"other_id":"n-728fca7147e339bb","other":{"id":"n-728fca7147e339bb","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-29d09fbfedb73813","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["23","12"],["385","552","144"]],"card_rows":[["1"],["23","12"],["385","552","144"],["5687","13860","9936","1728"]],"row_sums_preview":["1","35","1081","31211","863857","23226803","611452585","15839840123"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["D+1"],"edges":["e-ed9d8447019c9d1f"],"text":"A038326 \u2192 D+1"},"headline":"A038326 \u2192 D+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:           385,            552,             144;\n  1:          5687,          13860,            9936,            1728;\n  2:         78529,         272976,          332640,          158976,            20736;\n  3:       1039511,        4711740,         8189280,         6652800,          2384640,           248832;\n  4:      13367233,       74844792,       169622640,       196542720,        119750400,         34338816,          2985984;\n  5:     168298295,     1122847572,      3143481264,      4749433920,       4127397120,       2011806720,        480743424,        35831808;\n  6:    2085127297,    16156636320,     53896683456,    100591400448,     113986414080,      79246024704,      32188907520,      6593052672,       429981696;\n  7:   25508706839,   225193748076,    872458361280,   1940280604416,    2715967812096,    2462106544128,    1426428444672,    496628858880,     89006211072,      5159780352;\n  8:  308891147521,  3061044820680,  13511624884560,  34898334451200,   58208418132480,   65183227490304,   49242130882560,  24453059051520,   7449432883200,   1186749480960,    61917364224;\n  9: 3709051717943, 40773631472772, 202028958164880, 594511494920640, 1151645036889600, 1536702238697472, 1434031004786688, 928565896642560, 403475474350080, 109258348953600, 15665093148672, 743008370688;\n","recurrence_sentence":"The row polynomials satisfy (12*n+23+12*x)*P(n,x) = (132*n+385+(144*n+552)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 385 + 552*x + 144*x^2.","mathematica":"P[0, x_] := 385 + 552*x + 144*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((132*n + 385 + (144*n + 552)*x + 144*x^2)*P[n - 1, x])/(12*n + 23 + 12*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 385 + 552*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 385 + (144*n + 552)*x + 144*x^2)*v[i - 1])/(12*n + 23 + 12*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-108b34f10b7468f4","steps":["basis A013611","D+1"],"edges":["e-8f18b93eb5dfe7b3","e-92deea8cc97bba0a"],"text":"A038281 \u2192 basis A013611 \u2192 D+1"},"headline":"A038281 \u2192 basis A013611 \u2192 D+1","chain":null}
