{"id":"n-a43d8d44a6eb5699","revision":0,"first_index":0,"fingerprint":"f6-d115b0993943a210","name":"theta_plus_one_squared 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":false,"rows":[["1"],["4","48","81"],["16","384","1944","3456","2025"],["64","2304","19440","69120","121500","104976","35721"],["256","12288","145152","774144","2268000","3919104","4000752","2239488","531441"],["1024","61440","933120","6635520","27216000","70543872","120022560","134369280","95659380","39366000","7144929"],["4096","294912","5474304","48660480","256608000","886837248","2112397056","3547348992","4209012720","3464208000","1886261256","612220032","89813529"],["16384","1376256","30191616","322043904","2075673600","8966909952","27461161728","61487382528","102138708672","126097171200","114433182864","74282697216","32692124556","8749644624","1076168025"],["65536","6291456","159252480","1981808640","15095808000","78256668672","292919058432","819831767040","1750949291520","2882221056000","3661861851648","3565569466368","2615369964480","1399943139840","516560652000","117546246144","12440502369"],["262144","28311552","812187648","11551113216","101523456000","614013861888","2716158541824","9122491662336","23812910364672","48997757952000","80037837614592","103910881591296","106707094550784","85676520158208","52689186504000","23979434213376","7613587449828","1506290861232","139858796529"],["1048576","125829120","4034396160","64550338560","642981888000","4444290809856","22684401008640","88885816197120","274209270865920","677059928064000","1351750146379776","2193674166927360","2896335423521280","3100674062868480","2669585449536000","1822437000216576","964387743644880","381593684845440","106292685362040","18596183472000","1537671920841"],["4194304","553648128","19619905536","348798320640","3883106304000","30195034619904","174669887766528","782195182534656","2784278750330880","8020556070912000","18924502049316864","36853726004379648","59471420696303616","79583967613624320","88096319834688000","80187228009529344","59406285008524608","35259256479718656","16369073545754160","5727624509376000","1420808854857084","222763681811088","16600580533161"],["16777216","2415919104","93767860224","1833682599936","22562891366400","194943381405696","1260363111727104","6349584422928384","25615364503044096","84330418117017600","229589563323580416","521621968061988864","994794673465442304","1597467277189840896","2161296379944345600","2459074992292233216","2342304951764684544","1853629483505209344","1204763812967506176","632329745835110400","261428829293703456","81977034906480384","18327040908609744","2602870608208896","176518460300625"],["67108864","10468982784","441660211200","9422084505600","126977310720000","1206792361082880","8623537080238080","48271694443315200","217646887934361600","806099584942080000","2487220269338787840","6458176747434147840","14211352478077747200","26624454619830681600","42570989301934080000","58123590726907330560","67666587495424220160","66936620237688115200","55935462744919929600","39144222361221120000","22657165205454299520","10657014537842449920","3970858863532111200","1127910596890521600","229473998390812500","29783889199140336","1853020188851841"],["268435456","45097156608","2054739197952","47487305908224","695614832640000","7212134584811520","56444969979740160","347556199991869440","1732011023982919680","7127617382645760000","24579588544053903360","71799729721473761280","179063041223779614720","383392146525561815040","707333360709058560000","1126703451013895946240","1550180004440627589120","1840148541806989639680","1879431548229309634560","1644057339171287040000","1223486921094532174080","767305046724656394240","400262573444036808960","170540082249846865920","57827447594484750000","15011080156366729344","2801766525543983592","334794231980869248","19239382454622201"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","133","7825","353125","13890625","501953125","17119140625","559814453125","17730712890625","547637939453125","16574859619140625","493335723876953125","14479160308837890625","419914722442626953125","12053549289703369140625"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["4","-24","9"],["0","36","-36"],["0","0","36"]]},"terms":[{"coeff":{"coeffs":[["16","96","216","216","81"],["0","144","576","756","324"],["0","0","144","432","324"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191309,"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-693fa493adf4b12c"]},{"id":191310,"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-693fa493adf4b12c"]},{"id":191311,"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-693fa493adf4b12c"]}],"region_exact":{},"incoming_edges":[{"id":"e-693fa493adf4b12c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a43d8d44a6eb5699","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"f1dcfb71f7a2371cd51d61f17c47c2d9af6fc56b5a929b818dfa331cce14739b","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-ea068e5dc433974c","kind":"fingerprint_related","node_a_id":"n-a43d8d44a6eb5699","node_b_id":"n-66407c1a4552c33b","fingerprint":"f6-d115b0993943a210","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d115b0993943a210"},"other_id":"n-66407c1a4552c33b","other":{"id":"n-66407c1a4552c33b","name":"theta_plus_one_squared 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-d115b0993943a210","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","96","162"],["64","1536","7776","13824","8100"]],"card_rows":[["1"],["8","96","162"],["64","1536","7776","13824","8100"],["512","18432","155520","552960","972000","839808","285768"]],"row_sums_preview":["1","266","31300","2825000","222250000","16062500000","1095625000000","71656250000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-a142fb8711dcbce4","e-9fe5d9f1b7f869f1"],"text":"A139548 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A139548 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","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:      4,       48,        81;\n  2:     16,      384,      1944,        3456,         2025;\n  3:     64,     2304,     19440,       69120,       121500,       104976,         35721;\n  4:    256,    12288,    145152,      774144,      2268000,      3919104,       4000752,       2239488,         531441;\n  5:   1024,    61440,    933120,     6635520,     27216000,     70543872,     120022560,     134369280,       95659380,       39366000,        7144929;\n  6:   4096,   294912,   5474304,    48660480,    256608000,    886837248,    2112397056,    3547348992,     4209012720,     3464208000,     1886261256,       612220032,        89813529;\n  7:  16384,  1376256,  30191616,   322043904,   2075673600,   8966909952,   27461161728,   61487382528,   102138708672,   126097171200,   114433182864,     74282697216,     32692124556,     8749644624,     1076168025;\n  8:  65536,  6291456, 159252480,  1981808640,  15095808000,  78256668672,  292919058432,  819831767040,  1750949291520,  2882221056000,  3661861851648,   3565569466368,   2615369964480,  1399943139840,   516560652000,   117546246144,   12440502369;\n  9: 262144, 28311552, 812187648, 11551113216, 101523456000, 614013861888, 2716158541824, 9122491662336, 23812910364672, 48997757952000, 80037837614592, 103910881591296, 106707094550784, 85676520158208, 52689186504000, 23979434213376, 7613587449828, 1506290861232, 139858796529;\n","recurrence_sentence":"The row polynomials satisfy (4+(36*n-24)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (16+(144*n+96)*x+(144*n^2+576*n+216)*x^2+(432*n^2+756*n+216)*x^3+(324*n^2+324*n+81)*x^4)*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[((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*x^4)*P[n - 1, x])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)]];\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] = ((16 + (144*n + 96)*x + (144*n^2 + 576*n + 216)*x^2 + (432*n^2 + 756*n + 216)*x^3 + (324*n^2 + 324*n + 81)*x^4)*v[i - 1])/(4 + (36*n - 24)*x + (36*n^2 - 36*n + 9)*x^2)); 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","(\u03b8+1)\u00b2"],"edges":["e-cb7f9b19b94bd36e","e-693fa493adf4b12c"],"text":"A034870 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A034870 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}