{"id":"n-89760656b7074688","revision":0,"first_index":0,"fingerprint":"f6-d5fd215a74ece842","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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":[["315","108","9"],["5400","2835","486","27"],["91125","64800","17010","1944","81"],["1518750","1366875","486000","85050","7290","243"],["25059375","27337500","12301875","2916000","382725","26244","729"],["410062500","526246875","287043750","86113125","15309000","1607445","91854","2187"],["6663515625","9841500000","6314962500","2296350000","516678750","73483200","6429780","314928","6561"],["107641406250","179914921875","132860250000","56834662500","15500362500","2790065250","330674400","24800580","1062882","19683"],["1729951171875","3229242187500","2698723828125","1328602500000","426259968750","93002175000","13950326250","1417176000","93002175","3542940","59049"],["27679218750000","57088388671875","53282496093750","29685962109375","10960970625000","2813315793750","511511962500","65765823750","5845851000","341007975","11691702","177147"],["441137548828125","996451875000000","1027590996093750","639389953125000","267173658984375","78918988500000","16879894762500","2630632950000","295946206875","23383404000","1227628710","38263752","531441"],["7006302246093750","17204364404296875","19430811562500000","13358682949218750","6234052042968750","2083954540078125","512973425250000","94045127962500","12824335631250","1282433563125","91195275600","4352501790","124357194","1594323"],["110933118896484375","294264694335937500","361291652490234375","272031361875000000","140266170966796875","52366037160937500","14587681780546875","3077840551500000","493736921803125","59846899612500","5386220965125","348200143200","15233756265","401769396","4782969"],["1751575561523437500","4991990350341796875","6620955622558593750","5419374787353515625","3060352821093750000","1262395538701171875","392745278707031250","93777954303515625","17312853102187500","2468684609015625","269311048256250","22034540311875","1305750537000","52732233225","1291401630","14348907"],["27587315093994140625","84075626953125000000","119807768408203125000","105935289960937500000","65032497448242187500","29379387082500000000","10099164309609375000","2693110482562500000","562667725821093750","92335216545000000","11849686123275000","1175175483300000","88138161247500","4821232752000","180796228200","4132485216","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["432","8748","174960","3464208","68024448","1326476736","25713241344","495898225920","9521245937664","182093828557824","3470494144278528","65939388741292032","1249377891940270080","23613242157671104512","445278280687512256512"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["315","108","9"]],"recurrence":{"denominator":{"coeffs":[["6","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["105","36","3"],["15","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193067,"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-afacacf9c9f0cc39"]},{"id":193068,"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-afacacf9c9f0cc39"]},{"id":193069,"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-afacacf9c9f0cc39"]},{"id":193070,"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-afacacf9c9f0cc39"]}],"region_exact":{},"incoming_edges":[{"id":"e-afacacf9c9f0cc39","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-89760656b7074688","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":"670d7e16bed3fa8acf281f63b883d00645407ab70993df35759ca25b3b65dce7","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-8447c9ce68adee7a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-64bad4362972e752","kind":"fingerprint_related","node_a_id":"n-b939bab20b7b3344","node_b_id":"n-89760656b7074688","fingerprint":"f6-d5fd215a74ece842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d5fd215a74ece842"},"other_id":"n-b939bab20b7b3344","other":{"id":"n-b939bab20b7b3344","name":"one_plus_derivative of Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","first_index":0,"fingerprint":"f6-d5fd215a74ece842","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","1"],["35","12","1"]],"card_rows":[["1"],["6","1"],["35","12","1"],["200","105","18","1"]],"row_sums_preview":["1","7","48","324","2160","14256","93312","606528"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["D+1"],"edges":["e-6330884d385446ea"],"text":"A038243 \u2192 D+1"},"headline":"A038243 \u2192 D+1","chain":null}},{"id":"l-6a8aa7bb26ea853d","kind":"fingerprint_related","node_a_id":"n-fc60b80193fbbe49","node_b_id":"n-89760656b7074688","fingerprint":"f6-d5fd215a74ece842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d5fd215a74ece842"},"other_id":"n-fc60b80193fbbe49","other":{"id":"n-fc60b80193fbbe49","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-d5fd215a74ece842","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","2"],["140","48","4"]],"card_rows":[["1"],["12","2"],["140","48","4"],["1600","840","144","8"]],"row_sums_preview":["1","14","192","2592","34560","456192","5971968","77635584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","D+1"],"edges":["e-88ca34e9877f4f47","e-5c86206763c6dc3a"],"text":"A038291 \u2192 basis A013609 \u2192 D+1"},"headline":"A038291 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-cb5702e0fe56b7ca","kind":"fingerprint_related","node_a_id":"n-159ea4af14a6eac8","node_b_id":"n-89760656b7074688","fingerprint":"f6-d5fd215a74ece842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d5fd215a74ece842"},"other_id":"n-159ea4af14a6eac8","other":{"id":"n-159ea4af14a6eac8","name":"one_plus_derivative of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-d5fd215a74ece842","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","3"],["315","108","9"]],"card_rows":[["1"],["18","3"],["315","108","9"],["5400","2835","486","27"]],"row_sums_preview":["1","21","432","8748","174960","3464208","68024448","1326476736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","D+1"],"edges":["e-011a72a323f5e83e","e-07c98a3be6589550"],"text":"A147716 \u2192 basis A013610 \u2192 D+1"},"headline":"A147716 \u2192 basis A013610 \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:            315,            108,              9;\n  1:           5400,           2835,            486,             27;\n  2:          91125,          64800,          17010,           1944,             81;\n  3:        1518750,        1366875,         486000,          85050,           7290,           243;\n  4:       25059375,       27337500,       12301875,        2916000,         382725,         26244,          729;\n  5:      410062500,      526246875,      287043750,       86113125,       15309000,       1607445,        91854,        2187;\n  6:     6663515625,     9841500000,     6314962500,     2296350000,      516678750,      73483200,      6429780,      314928,       6561;\n  7:   107641406250,   179914921875,   132860250000,    56834662500,    15500362500,    2790065250,    330674400,    24800580,    1062882,     19683;\n  8:  1729951171875,  3229242187500,  2698723828125,  1328602500000,   426259968750,   93002175000,  13950326250,  1417176000,   93002175,   3542940,    59049;\n  9: 27679218750000, 57088388671875, 53282496093750, 29685962109375, 10960970625000, 2813315793750, 511511962500, 65765823750, 5845851000, 341007975, 11691702, 177147;\n","recurrence_sentence":"The row polynomials satisfy (n+6+x)*P(n,x) = (15*n+105+(3*n+36)*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 315 + 108*x + 9*x^2.","mathematica":"P[0, x_] := 315 + 108*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n + 105 + (3*n + 36)*x + 3*x^2)*P[n - 1, x])/(n + 6 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 315 + 108*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15*n + 105 + (3*n + 36)*x + 3*x^2)*v[i - 1])/(n + 6 + 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-615df6ea54d3bbef","steps":["x \u21a6 x+1","D+1"],"edges":["e-5083e5230277d126","e-afacacf9c9f0cc39"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}