{"id":"n-b856b70338e77df7","revision":0,"first_index":0,"fingerprint":"f6-484d946563c33157","name":"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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["12","18"],["96","432","648","324"],["576","4320","12960","19440","14580","4374"],["3072","32256","145152","362880","544320","489888","244944","52488"],["15360","207360","1244160","4354560","9797760","14696640","14696640","9447840","3542940","590490"],["73728","1216512","9123840","41057280","123171840","258660864","387991296","415704960","311778720","155889360","46766808","6377292"],["344064","6709248","60383232","332107776","1245404160","3362591232","6725182464","10087773696","11348745408","9457287840","5674372704","2321334288","580333572","66961566"],["1572864","35389440","371589120","2415329280","10868981760","35867639808","89669099520","172933263360","259399895040","302633210880","272369889792","185706743040","92853371520","32141551680","6887475360","688747536"],["7077888","180486144","2165833728","16243752960","85279703040","332590841856","997772525568","2351892381696","4409798215680","6614697323520","7937636788224","7576835116032","5682626337024","3278438271360","1405044973440","421513492032","79033779756","6973568802"],["31457280","896532480","12103188480","102877102080","617262612480","2777681756160","9721886146560","27082397122560","60935393525760","111714888130560","167572332195840","205656953149440","205656953149440","166107539082240","106783417981440","53391708990720","20021890871520","5299912289520","883318714920","69735688020"],["138412032","4359979008","65399685120","621297008640","4193754808320","21388149522432","85552598089728","274990493859840","721850046382080","1564008433827840","2815215180890112","4222822771335168","5278528464168960","5481548789713920","4698470391183360","3288929273828352","1850022716528448","816186492586080","272062164195360","64435775730480","9665366359572","690383311398"],["603979776","20837302272","343815487488","3610062618624","27075469639680","154330176946176","694485796257792","2529912543510528","7589737630531584","18974344076328960","39846122560290816","70636308175060992","105954462262591488","134480663640981504","144086425329623040","129677782796660736","97258337097495552","60071325854335488","30035662927167744","11856182734408320","3556854820322496","762183175783392","103934069425008","6778308875544"],["2617245696","98146713600","1766640844800","20316369715200","167610050150400","1055943315947520","5279716579737600","21495988931788800","72548962644787200","205555394160230400","493332945984552960","1009090116786585600","1765907704376524800","2648861556564787200","3405679144154726400","3746247058570199040","3512106617409561600","2789025843237004800","1859350562158003200","1027535836982054400","462391126641924480","165139688086401600","45038096750836800","8811801538207200","1101475192275900","66088511536554"],["11274289152","456608710656","8903869857792","111298373222400","1001685359001600","6911628977111040","38013959374110720","171062817183498240","641485564438118400","2031370954054041600","5484701575945912320","12714535471510978560","25429070943021957120","44011853555230310400","66017780332845465600","85823114432699105280","96551003736786493440","93711268332763361280","78092723610636134400","55486935197030937600","33292161118218562560","16646080559109281280","6809760228726524160","2220573987628214400","555143496907053600","99925829443269648","11529903397300344","640550188738908"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","30","1500","56250","1875000","58593750","1757812500","51269531250","1464843750000","41198730468750","1144409179687500","31471252441406250","858306884765625000","23245811462402343750","625848770141601562500"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["12","18"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["4","12","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191300,"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-8cb133e98a7e7cc2"]},{"id":191301,"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-8cb133e98a7e7cc2"]},{"id":191302,"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-8cb133e98a7e7cc2"]}],"region_exact":{},"incoming_edges":[{"id":"e-8cb133e98a7e7cc2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-b856b70338e77df7","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":"5b179b5fb7525ecd9b838095d8245c8982d2ac952bc1f7f263f2be319532f7ac","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-76c438f0739bbf16","kind":"fingerprint_related","node_a_id":"n-b856b70338e77df7","node_b_id":"n-c9845261e64d60a9","fingerprint":"f6-484d946563c33157","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-484d946563c33157"},"other_id":"n-c9845261e64d60a9","other":{"id":"n-c9845261e64d60a9","name":"basis_map_A013610 of derivative of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-484d946563c33157","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4","6"],["32","144","216","108"]],"card_rows":[["4","6"],["32","144","216","108"],["192","1440","4320","6480","4860","1458"],["1024","10752","48384","120960","181440","163296","81648","17496"]],"row_sums_preview":["0","10","500","18750","625000","19531250","585937500","17089843750"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["D","basis A013610"],"edges":["e-473cc94657c879cc","e-32720d39f4766d5d"],"text":"A034870 \u2192 D \u2192 basis A013610"},"headline":"A034870 \u2192 D \u2192 basis A013610","chain":null}},{"id":"l-e09e1c1d036d2a28","kind":"fingerprint_related","node_a_id":"n-b856b70338e77df7","node_b_id":"n-0e81ff2ad5810903","fingerprint":"f6-484d946563c33157","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-484d946563c33157"},"other_id":"n-0e81ff2ad5810903","other":{"id":"n-0e81ff2ad5810903","name":"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":"f6-484d946563c33157","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["24","36"],["384","1728","2592","1296"]],"card_rows":[["24","36"],["384","1728","2592","1296"],["4608","34560","103680","155520","116640","34992"],["49152","516096","2322432","5806080","8709120","7838208","3919104","839808"]],"row_sums_preview":["0","60","6000","450000","30000000","1875000000","112500000000","6562500000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013610","D"],"edges":["e-a142fb8711dcbce4","e-ebf0ead7cd3d66b0"],"text":"A139548 \u2192 basis A013610 \u2192 D"},"headline":"A139548 \u2192 basis A013610 \u2192 D","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:       0;\n  1:      12,        18;\n  2:      96,       432,        648,         324;\n  3:     576,      4320,      12960,       19440,       14580,         4374;\n  4:    3072,     32256,     145152,      362880,      544320,       489888,       244944,         52488;\n  5:   15360,    207360,    1244160,     4354560,     9797760,     14696640,     14696640,       9447840,       3542940,        590490;\n  6:   73728,   1216512,    9123840,    41057280,   123171840,    258660864,    387991296,     415704960,     311778720,     155889360,      46766808,       6377292;\n  7:  344064,   6709248,   60383232,   332107776,  1245404160,   3362591232,   6725182464,   10087773696,   11348745408,    9457287840,    5674372704,    2321334288,     580333572,      66961566;\n  8: 1572864,  35389440,  371589120,  2415329280, 10868981760,  35867639808,  89669099520,  172933263360,  259399895040,  302633210880,  272369889792,  185706743040,   92853371520,   32141551680,    6887475360,    688747536;\n  9: 7077888, 180486144, 2165833728, 16243752960, 85279703040, 332590841856, 997772525568, 2351892381696, 4409798215680, 6614697323520, 7937636788224, 7576835116032, 5682626337024, 3278438271360, 1405044973440, 421513492032, 79033779756, 6973568802;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n+12*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12 + 18*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 12 + 18*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 12*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 12 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 12*n*x + 9*n*x^2)*v[i - 1])/(n - 1)); 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"],"edges":["e-cb7f9b19b94bd36e","e-8cb133e98a7e7cc2"],"text":"A034870 \u2192 basis A013610 \u2192 D"},"headline":"A034870 \u2192 basis A013610 \u2192 D","chain":null}
