{"id":"n-ef2d25c676dbb756","revision":0,"first_index":0,"fingerprint":"f6-7939c1a278d41502","name":"one_plus_derivative of basis_map_A013610 of Even-numbered rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["16","30","9"],["112","528","864","540","81"],["640","4896","15120","23760","19440","7290","729"],["3328","35328","161280","411264","635040","598752","326592","87480","6561"],["16384","222720","1347840","4769280","10886400","16656192","17146080","11547360","4723920","984150","59049"],["77824","1290240","9732096","44098560","133436160","283295232","431101440","471132288","363741840","190531440","62355744","10628820","531441"],["360448","7053312","63737856","352235520","1328431104","3611672064","7285614336","11048514048","12609717120","10718259552","6620101488","2837186352","773778096","111602610","4782969"],["1638400","36962304","389283840","2539192320","11472814080","38041436160","95647039488","185743134720","281016552960","331455421440","302633210880","210467642112","108328933440","39284118720","9183300480","1147912560","43046721"],["7340032","187564032","2256076800","16965697536","89340641280","349646782464","1053204332544","2494431313920","4703784763392","7104674903040","8599106520576","8298438460416","6314029263360","3715563374208","1639219135680","515183156928","105378373008","11622614670","387420489"],["32505856","927989760","12551454720","106911498240","642981888000","2901134278656","10184833105920","28471238000640","64320693166080","118485487411200","178743821008896","220890801530880","222795032578560","181927304709120","118648242201600","60510603522816","23358872683440","6477670576080","1177758286560","116226146700","3486784401"],["142606336","4498391040","67579674624","643096903680","4349079060480","22226900484096","89117289676800","287212293586944","756223858114560","1644213994536960","2971616024272896","4478751424143360","5630430361780224","5887589440803840","5090009590448640","3602160633240576","2055580796142720","925011358264224","317405858227920","78754837003920","12887155146096","1150638852330","31381059609"],["620756992","21441282048","354234138624","3724667781120","27977985294336","159745270874112","720207492415488","2629124800118784","7905976698470400","19817648257499136","41743556967923712","74258682953269248","111840821277179904","142631006891950080","153692187018264576","139283544485302272","105363198522286848","65792404507129344","33372958807964160","13437007098996096","4149663957042912","931557214846368","138578759233344","11297181459240","282429536481"],["2684354560","100763959296","1815714201600","20905249996800","172689142579200","1089465325977600","5455707132395520","22250234157465600","75235961261260800","213616390009651200","513888485400576000","1053938566421544960","1849998547442073600","2784700610747596800","3594883541052211200","3973292334847180800","3746247058570199040","2995620350143449600","2014296442337836800","1125396392885107200","513767918491027200","187158313164588480","52544446209309600","10769979657808800","1468633589701200","110147519227590","2541865828329"],["11542724608","467882999808","9132174213120","114266329841664","1029509952307200","7111966048911360","39165897536962560","176493382808371200","662868416586055680","2102647127880499200","5687838671351316480","13213144705687879680","26488615565647872000","45967935935462768640","69161484158219059200","90224299788222136320","101914948388830187520","99390739140809625600","83298905184678543360","59597078544959155200","36066507878070109440","18231421564738736640","7566400254140582400","2516650519311976320","647667413058229200","122131569319551792","15373204529733792","1067583647898180","22876792454961"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","55","2125","71875","2265625","68359375","2001953125","57373046875","1617431640625","45013427734375","1239776611328125","33855438232421875","917911529541015625","24735927581787109375","663101673126220703125"],"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":[["-4","3"],["6","0"]]},"terms":[{"coeff":{"coeffs":[["8","36","54","27"],["24","72","54","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191303,"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-29a2647da7edee02"]},{"id":191304,"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-29a2647da7edee02"]},{"id":191305,"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-29a2647da7edee02"]}],"region_exact":{},"incoming_edges":[{"id":"e-29a2647da7edee02","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-ef2d25c676dbb756","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":"fadaf75c169704c355fcc8bd189b014ae93c329dac4d69a4e97eb64a087228ed","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-1e083728b424e77f","role":"sequence","name":"basis_map_A013610 of Even-numbered rows of Pascal's triangle.","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-eea811b88144cd49","kind":"fingerprint_related","node_a_id":"n-ef2d25c676dbb756","node_b_id":"n-c5c61f28a5dc4bc9","fingerprint":"f6-7939c1a278d41502","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7939c1a278d41502"},"other_id":"n-c5c61f28a5dc4bc9","other":{"id":"n-c5c61f28a5dc4bc9","name":"one_plus_derivative of basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-7939c1a278d41502","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["32","60","18"],["448","2112","3456","2160","324"]],"card_rows":[["1"],["32","60","18"],["448","2112","3456","2160","324"],["5120","39168","120960","190080","155520","58320","5832"]],"row_sums_preview":["1","110","8500","575000","36250000","2187500000","128125000000","7343750000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013610","D+1"],"edges":["e-a142fb8711dcbce4","e-5bbb999a3b790205"],"text":"A139548 \u2192 basis A013610 \u2192 D+1"},"headline":"A139548 \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:       1;\n  1:      16,        30,          9;\n  2:     112,       528,        864,         540,          81;\n  3:     640,      4896,      15120,       23760,       19440,         7290,           729;\n  4:    3328,     35328,     161280,      411264,      635040,       598752,        326592,         87480,          6561;\n  5:   16384,    222720,    1347840,     4769280,    10886400,     16656192,      17146080,      11547360,       4723920,        984150,         59049;\n  6:   77824,   1290240,    9732096,    44098560,   133436160,    283295232,     431101440,     471132288,     363741840,     190531440,      62355744,      10628820,        531441;\n  7:  360448,   7053312,   63737856,   352235520,  1328431104,   3611672064,    7285614336,   11048514048,   12609717120,   10718259552,    6620101488,    2837186352,     773778096,     111602610,       4782969;\n  8: 1638400,  36962304,  389283840,  2539192320, 11472814080,  38041436160,   95647039488,  185743134720,  281016552960,  331455421440,  302633210880,  210467642112,  108328933440,   39284118720,    9183300480,   1147912560,     43046721;\n  9: 7340032, 187564032, 2256076800, 16965697536, 89340641280, 349646782464, 1053204332544, 2494431313920, 4703784763392, 7104674903040, 8599106520576, 8298438460416, 6314029263360, 3715563374208, 1639219135680, 515183156928, 105378373008, 11622614670, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (6*n-4+3*x)*P(n,x) = (24*n+8+(72*n+36)*x+(54*n+54)*x^2+27*x^3)*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[((24*n + 8 + (72*n + 36)*x + (54*n + 54)*x^2 + 27*x^3)*P[n - 1, x])/(6*n - 4 + 3*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] = ((24*n + 8 + (72*n + 36)*x + (54*n + 54)*x^2 + 27*x^3)*v[i - 1])/(6*n - 4 + 3*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-08f49539099949d5","steps":["basis A013610","D+1"],"edges":["e-cb7f9b19b94bd36e","e-29a2647da7edee02"],"text":"A034870 \u2192 basis A013610 \u2192 D+1"},"headline":"A034870 \u2192 basis A013610 \u2192 D+1","chain":null}
